<table><caption>**Table 5.** HSE-to-lensing mass bias values from resolved mass profiles.</caption><thead><tr><th>Reference</th><th>Sample</th><th># of clusters</th><th>Redshift</th><th>(1 - b) = M<sup>HSE</sup><sub>500</sub>/M<sup>lens</sup><sub>500</sub></th><th>Notes</th></tr></thead><tbody><tr><td></td><td>This work – Reference SR</td><td>53</td><td>0.05 &lt; z &lt; 1.07</td><td>0.739<sup>+0.075</sup><sub>-0.070</sub></td><td>Propagating the systematic uncertainties, accounting for intrinsic dispersion</td></tr><tr><td>I</td><td>Smith et al. (2015)</td><td>50</td><td>0.15 &lt; z &lt; 0.3</td><td>0.95 ± 0.05</td><td>Weighted mean</td></tr><tr><td>II</td><td>Mahdavi et al. (2008)</td><td>18</td><td>0.170 &lt; z &lt; 0.547</td><td>0.78 ± 0.09</td><td>M<sup>HSE</sup>(R<sup>lens</sup><sub>500</sub>)</td></tr><tr><td>III</td><td>Mahdavi et al. (2013)</td><td>50</td><td>0.152 &lt; z &lt; 0.55</td><td>0.8–1</td><td>M<sup>HSE</sup>(R<sup>lens</sup><sub>500</sub>)</td></tr><tr><td>IV</td><td>Israel et al. (2014)</td><td>8</td><td>0.35 &lt; z &lt; 0.80</td><td>0.8–1</td><td>Global temperature profile for the whole sample, M<sup>HSE</sup>(R<sup>lens</sup><sub>500</sub>)</td></tr><tr><td>V</td><td>Bartalucci et al. (2018)</td><td>4</td><td>0.933 &lt; z &lt; 1.066</td><td>1.39<sup>+0.51</sup><sub>-0.37</sub></td><td>Weighted mean</td></tr><tr><td>VI</td><td>Eckert et al. (2022)</td><td>12</td><td>0.047 &lt; z &lt; 0.09</td><td>0.85–0.9</td><td></td></tr><tr><td>VII</td><td>Lovisari et al. (2020a)</td><td>62</td><td>z &lt; 0.5</td><td>0.74 ± 0.06</td><td>Accounting for intrinsic dispersion</td></tr><tr><td>VIII</td><td>Sereno et al. (2019)</td><td>100</td><td>0.054 &lt; z &lt; 1.050</td><td>0.91 ± 0.17</td><td>Accounting for intrinsic dispersion, temperature measured within 300 kpc</td></tr></tbody></table>

**Notes.** We report our reference result and different values from the literature. The last column indicates the singularity of each analysis.

$z = 0.30$, spanning from $z = 0.054$ to $z = 1.050$. Thus, the analysis in Sereno et al. (2019) is probably the closest study to our work. Nevertheless, to get temperature profiles that reach $R_{500}$ with X-ray data (to compute then the HSE mass), in Sereno et al. (2019) a model was iteratively fitted to the integrated temperature measured per cluster within 300 kpc, well below $R_{500}$. Assuming $\beta^{HSE} = 1$, $\beta^{lens} = 1$, and $\alpha^{lens} = 0$ they obtained: $\alpha^{HSE} = -0.04 \pm 0.08$, $\sigma^{HSE} = 0.31 \pm 0.05$, and $\sigma^{lens} = 0.37 \pm 0.06$. According to Sereno et al. (2019), the difference between HSE and lensing masses is of $b = 0.09 \pm 0.17$ (VIII). The $\alpha^{HSE}$ from Sereno et al. (2019) is at $3\sigma$ from our result with the full reference sample ($\alpha^{HSE} = -0.338^{+0.105}_{-0.097}$ without accounting for the systematic uncertainties). Their values for $\sigma^{HSE}$ and $\sigma^{lens}$ agree with the intrinsic scatter values that we obtain when we do not account for the systematic uncertainties ($\sigma^{HSE} = 0.304^{+0.069}_{-0.072}$ and $\sigma^{lens} = 0.305^{+0.080}_{-0.083}$).

In addition, the behaviour of the HSE-to-lensing mass bias could vary with the overdensity at which masses are measured. By estimating weak lensing masses and HSE masses from X-rays at $R_{200}$, Jee et al. (2011) concluded that for a sample of 14 very massive and distant clusters ($0.83 < z < 1.46$), the HSE and lensing masses are compatible. However, the HSE masses were obtained from the extrapolation of a singular isothermal sphere profile to reach $R_{200}$, which likely limits the validity of their HSE mass estimates. Similarly, in Amodeo et al. (2016) authors compared $M_{200}$ masses reconstructed from Chandra data (although the radial reach of Chandra is way below $R_{200}$) to their lensing estimates, and concluded that both mass estimates are in agreement. No evolution with redshift was detected in Amodeo et al. (2016). We prefer to avoid extrapolating the mass profiles to reach $R_{200}$.

## 7. Summary and conclusions

In this work we have investigated the HSE-to-lensing mass bias with masses inferred at $R_{500}$ from resolved profiles. We carefully selected the clusters and obtained a reference sample with 53 clusters with redshifts spanning from $z = 0.05$ to $1.07$. This is the largest redshift range analysed homogeneously with this type of data, having access to X-ray HSE masses obtained from resolved profiles. HSE masses were estimated with the XMM-Newton mass reconstruction reference pipeline and lensing masses were extracted from the $LC^2$ CoMaLit catalogue.

In order to account for possible systematic effects in the reference analysis, we compared the XMM-Newton and CoMaLit masses to other estimates from the literature. The obtained systematic scatters were propagated to our analyses, but all the main conclusions remain unchanged when considering or not these additional systematic dispersions on the HSE and lensing mass uncertainties.

We performed different tests in the measurement of the HSE-to-lensing mass scaling relation and bias, varying the redshift range and the scaling relation model. Our main conclusions are the following:

1. Assuming that HSE and lensing masses scale linearly with the true mass and considering $\sigma^2_{sys HSE}$ and $\sigma^2_{sys lens}$, we measure for the 53 clusters in the reference sample a HSE-to-lensing mass ratio of $M_{500}^{HSE}/M_{500}^{lens} = (1 - b) = 0.739^{+0.075}_{-0.070}$ (stat.) $\pm 0.226$ (intrinsic scatter).

2. We find that the best scaling relation between HSE and lensing masses is our scaling relation of reference, where we assume that there is no evolution with mass and redshift and that HSE and lensing masses scale linearly. We obtain: $\alpha^{HSE} = -0.303^{+0.101}_{-0.095}$, $\sigma^{HSE} = 0.166^{+0.086}_{-0.101}$, and $\sigma^{lens} = 0.257^{+0.080}_{-0.092}$.

3. When we let the SR evolve with redshift, we observe a trend towards a larger discrepancy between HSE and lensing masses at high redshift, but it is not statistically significant. In conclusion, there is no evidence of evolution with redshift. The dependence of the HSE-to-lensing mass bias on the mass of the clusters is not confirmed either.

4. Given the size of the sample, single clusters can be driving the fits and special care needs to be taken for clusters with very small uncertainties. We have investigated the case of CL J1226.9+3332 galaxy cluster, whose impact is crucial when determining the bias at high redshift.

5. Ignoring the intrinsic scatter of HSE and lensing masses with respect to the true mass of clusters introduces a bias in the measurement of the HSE-to-lensing mass bias.

Additional considerations are needed to compare the HSE-to-lensing mass bias obtained in this work to the bias needed to reconcile cluster number counts and CMB power spectrum results for several reasons: (1) the HSE masses used in cluster number count analyses are not direct HSE mass measurements, but masses obtained from a SZ (or X-ray) measurement through a SZ-mass (or X-ray-mass) scaling relation, (2) lensing masses
can also be biased with respect to the true mass of clusters (Becker & Kravtsov 2011), and (3) this sample is not representative of the cluster population in any given survey. Instead, this study provide a step forwards in our understanding of the deviation from hydrostatic equilibrium of galaxy clusters and of the impact of systematic and intrinsic errors.

**Acknowledgements.** We thank M. Sereno and S. Ettori for very useful comments. This project was carried out using the Python libraries `matplotlib` (Hunter 2007), `numpy` (Harris et al. 2020), `astropy` (Astropy Collaboration 2013, 2018), `scipy` (Virtanen et al. 2020), and `ChainConsumer` (Hinton 2016). This work benefited the financial support from CNRS/INSU and CNES. A.F. also thanks financial support by Universidad de La Laguna (ULL), NextGenerationEU/PRTR and Ministerio de Universidades (MIU) (UNI/511/2021) through grant “Margarita Salas”.

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# Appendix A: Measuring the systematic scatter

We present in Fig. A.1 different measurements of the systematic scatter for HSE masses and lensing masses obtained from the reference analyses. Top panels do not account for the HSE

masses obtained by evaluating the HSE mass profiles at the $R_{500}$ from lensing. In the bottom panels in Fig. A.1 we present the systematic scatters once the cluster selection criteria from Sect. 3.3 have been applied. We summarise all the scatter values in Table A.1.

<figure><img src="image_15.png" /></figure>

**Fig. A.1.** Comparison of HSE and lensing mass estimates from the homogeneous samples in this work (XMM-Newton and CoMaLit) with respect to other estimates from the literature (comparison sample). Top: Relation between X-ray masses from literature and from the XMM-Newton reference pipeline without accounting for $M^{\text{HSE}}(< R_{500}^{\text{lens}})$. In the left (right) the clusters with very large uncertainties and with very different XMM-Newton and CoMaLit centres are considered (not considered). Bottom: Same figure as Fig. 1, but not accounting for clusters with very large uncertainties and with very different XMM-Newton and CoMaLit centres. The dashed lines show the 1:1 relation. We give the statistical, systematic, and raw variances in units of $10^{28} M_{\odot}^2$ corresponding to the data points in each figure.
# Appendix A: Measuring the systematic scatter

We present in Fig. A.1 different measurements of the systematic scatter for HSE masses and lensing masses obtained from the reference analyses. Top panels do not account for the HSE

masses obtained by evaluating the HSE mass profiles at the $R_{500}$ from lensing. In the bottom panels in Fig. A.1 we present the systematic scatters once the cluster selection criteria from Sect. 3.3 have been applied. We summarise all the scatter values in Table A.1.

<figure><img src="image_16.png" /></figure>

**Fig. A.1.** Comparison of HSE and lensing mass estimates from the homogeneous samples in this work (XMM-Newton and CoMaLit) with respect to other estimates from the literature (comparison sample). Top: Relation between X-ray masses from literature and from the XMM-Newton reference pipeline without accounting for $M^{\text{HSE}}(< R_{500}^{\text{lens}})$. In the left (right) the clusters with very large uncertainties and with very different XMM-Newton and CoMaLit centres are considered (not considered). Bottom: Same figure as Fig. 1, but not accounting for clusters with very large uncertainties and with very different XMM-Newton and CoMaLit centres. The dashed lines show the 1:1 relation. We give the statistical, systematic, and raw variances in units of $10^{28} M_{\odot}^2$ corresponding to the data points in each figure.
<table><caption>**Table A.1.** Raw, statistical, and systematic variances of the HSE and lensing mass estimates from the homogeneous samples in this work (XMM-Newton and CoMaLit) with respect to other estimates from the literature (comparison samples).</caption><thead><tr><th>Sample</th><th>σ<sup>2</sup><sub>raw HSE</sub><br>[10<sup>28</sup> M<sub>☉</sub><sup>2</sup>]</th><th>σ<sup>2</sup><sub>stat HSE</sub><br>[10<sup>28</sup> M<sub>☉</sub><sup>2</sup>]</th><th>σ<sup>2</sup><sub>sys HSE</sub><br>[10<sup>28</sup> M<sub>☉</sub><sup>2</sup>]</th><th>Figure</th></tr></thead><tbody><tr><td>All</td><td>3.231</td><td>1.507</td><td>1.724</td><td>Fig. 1</td></tr><tr><td>Without M<sup>HSE</sup>(<i>R</i><sub>500</sub><sup>lens</sup>)</td><td>3.393</td><td>1.501</td><td>1.892</td><td>Fig. A.1</td></tr><tr><td>Without clusters discarded in Sect. 3.3</td><td>3.161</td><td>1.396</td><td>1.765</td><td>Fig. 1</td></tr><tr><td>Without M<sup>HSE</sup>(<i>R</i><sub>500</sub><sup>lens</sup>) and clusters discarded in Sect. 3.3</td><td>3.405</td><td>1.388</td><td><b>2.017</b></td><td>Fig. A.1</td></tr><tr><td></td><td>σ<sup>2</sup><sub>raw lens</sub><br>[10<sup>28</sup> M<sub>☉</sub><sup>2</sup>]</td><td>σ<sup>2</sup><sub>stat lens</sub><br>[10<sup>28</sup> M<sub>☉</sub><sup>2</sup>]</td><td>σ<sup>2</sup><sub>sys lens</sub><br>[10<sup>28</sup> M<sub>☉</sub><sup>2</sup>]</td><td>Figure</td></tr><tr><td>All</td><td>5.280</td><td>4.340</td><td>0.940</td><td>Fig. 1</td></tr><tr><td>Without clusters discarded in Sect. 3.3</td><td>5.612</td><td>4.409</td><td><b>1.202</b></td><td>Fig. 1</td></tr></tbody></table>

**Notes.** We report the different values depending on the sample selection criteria, showing in bold the systematic scatters considered for the rest of the analysis.

## Appendix B: Additional checks for the selection of the reference sample

In this section we detail the additional checks we have performed in Sect. 3.3 to verify which clusters can be used for the HSE-lensing mass comparison and define the final sample.

### B.1. Uncertainties of mass estimates

Regarding the uncertainties of the mass estimates in the homogeneous sample (the 65 clusters in Sect. 3.3), we observe that masses and their uncertainties appear correlated, error bars being larger for more massive objects. As already mentioned, lensing mass uncertainties are larger than HSE ones and this is also true for relative uncertainties when calculated with respect to the value of the measured mass. For some clusters, the uncertainties on HSE masses are suspiciously large (larger than the value of the mass) or abnormal (negative error bars) and we decide to exclude for these reasons the clusters Abell119, Abell521 (the LoCuSS and Mahdavi+2013 cluster out of the plot in the left panel in Fig. 1), and SPT-CLJ0516-54.

### B.2. Difference of centres for HSE and lensing masses

For some of the clusters, the HSE and lensing masses have been reconstructed assuming different cluster centre positions. We investigate in the following if this miscentring is correlated to the mass ratio and redshift and how it may affect the mass estimates.

#### B.2.1. Correlation between miscentring and bias or redshift

In the left panel in Fig. B.1 we present the ratios between the HSE and lensing masses of the clusters with respect to the separations between the centres considered in the X-ray and lensing analyses. Each marker corresponds to one of the 65 clusters, showing in magenta crosses, purple squares, and grey circles the clusters from ESZ+LoCuSS, LPSZ, and Bartalucci+2018 samples, respectively. Error bars have been obtained from the propagation of the individual XMM-Newton and CoMaLit uncertainties. The separation between X-ray and lensing centres goes from 1.5 kpc to 700 kpc. However, there is no significative correlation between the miscentring and the HSE-to-lensing mass ratio.

The miscentring could also be related to the redshift of the cluster. In the right panel in Fig. B.1 we show the redshift with

respect to the separation of the centres. There is neither indication of correlation. We do not find any significative correlation between the redshift and the uncertainties of mass ratios either.

The dynamical state of clusters is an important point to better understand the evolution of the HSE bias. In particular, relaxed clusters tend to have smaller HSE bias as compared to disturbed ones (Gianfagna et al. 2021). The offset between the centres used in lensing and X-ray analyses could be an indicator of the departure from sphericity and equilibrium of clusters, resulting in a difficulty to define the centre. However, the absence of correlation between the separation of centres and the bias or the redshift does not enable any clear dynamical classification of the clusters in our sample.

#### B.2.2. Simulations to quantify the impact of miscentring

It is evident that two mass estimates computed assuming very different centres are hardly comparable. Nevertheless, setting a quantitative limit of the acceptable separation between the two centres can be difficult. In this section we make use of simulated mock mass density profiles to check the impact that miscentring has on the mass profiles and, consequently, on $M_{500}$.

We simulate mass density profiles using the profiley<fnref n="9" /> Python package, which contains already tested (Madhavacheril et al. 2020) functions that describe density profiles, as well as the corresponding miscentred density profiles. We test a variety of density profile shapes following NFW, gNFW, truncated Navarro-Frenk-White (tNFW, Baltz et al. 2009), and Hernquist (Hernquist 1990) models. To initialise the density profiles we consider different $M_{500}$ (in the range ~ 1 to $9 \times 10^{14}$ M☉) and model parameters. The parameters considered for each model are given in Table B.1.

For each density profile we build the corresponding miscentred profiles by displacing the centre by 0.0 to 0.7 Mpc from the original one. We integrate each miscentred density profile to get a mass profile and obtain the miscentred $M_{500}$ estimate at the radius where the overdensity reaches $\Delta = 500$, $M_{500}^{miscent}$. In Fig. B.2 we show the relative error of the miscentred masses with respect to the separation to the true centre, for all type of profiles and for different redshifts, $z = [0.1, 0.45, 0.8, 1.15, 1.5]$. Each marker indicates a different mass density model and the colours show the true $M_{500}$. For visualisation purposes gNFW, tNFW, and Hernquist markers have been shifted by 0.01, 0.02,

<footnote n="9">⁹ https://profiley.readthedocs.io/en/latest/index.html.</footnote>
<table><caption>**Table A.1.** Raw, statistical, and systematic variances of the HSE and lensing mass estimates from the homogeneous samples in this work (XMM-Newton and CoMaLit) with respect to other estimates from the literature (comparison samples).</caption><thead><tr><th>Sample</th><th>σ<sup>2</sup><sub>raw HSE</sub><br>[10<sup>28</sup> M<sub>☉</sub><sup>2</sup>]</th><th>σ<sup>2</sup><sub>stat HSE</sub><br>[10<sup>28</sup> M<sub>☉</sub><sup>2</sup>]</th><th>σ<sup>2</sup><sub>sys HSE</sub><br>[10<sup>28</sup> M<sub>☉</sub><sup>2</sup>]</th><th>Figure</th></tr></thead><tbody><tr><td>All</td><td>3.231</td><td>1.507</td><td>1.724</td><td>Fig. 1</td></tr><tr><td>Without M<sup>HSE</sup>(<i>R</i><sub>500</sub><sup>lens</sup>)</td><td>3.393</td><td>1.501</td><td>1.892</td><td>Fig. A.1</td></tr><tr><td>Without clusters discarded in Sect. 3.3</td><td>3.161</td><td>1.396</td><td>1.765</td><td>Fig. 1</td></tr><tr><td>Without M<sup>HSE</sup>(<i>R</i><sub>500</sub><sup>lens</sup>) and clusters discarded in Sect. 3.3</td><td>3.405</td><td>1.388</td><td><b>2.017</b></td><td>Fig. A.1</td></tr><tr><td></td><td>σ<sup>2</sup><sub>raw lens</sub><br>[10<sup>28</sup> M<sub>☉</sub><sup>2</sup>]</td><td>σ<sup>2</sup><sub>stat lens</sub><br>[10<sup>28</sup> M<sub>☉</sub><sup>2</sup>]</td><td>σ<sup>2</sup><sub>sys lens</sub><br>[10<sup>28</sup> M<sub>☉</sub><sup>2</sup>]</td><td>Figure</td></tr><tr><td>All</td><td>5.280</td><td>4.340</td><td>0.940</td><td>Fig. 1</td></tr><tr><td>Without clusters discarded in Sect. 3.3</td><td>5.612</td><td>4.409</td><td><b>1.202</b></td><td>Fig. 1</td></tr></tbody></table>

**Notes.** We report the different values depending on the sample selection criteria, showing in bold the systematic scatters considered for the rest of the analysis.

## Appendix B: Additional checks for the selection of the reference sample

In this section we detail the additional checks we have performed in Sect. 3.3 to verify which clusters can be used for the HSE-lensing mass comparison and define the final sample.

### B.1. Uncertainties of mass estimates

Regarding the uncertainties of the mass estimates in the homogeneous sample (the 65 clusters in Sect. 3.3), we observe that masses and their uncertainties appear correlated, error bars being larger for more massive objects. As already mentioned, lensing mass uncertainties are larger than HSE ones and this is also true for relative uncertainties when calculated with respect to the value of the measured mass. For some clusters, the uncertainties on HSE masses are suspiciously large (larger than the value of the mass) or abnormal (negative error bars) and we decide to exclude for these reasons the clusters Abell119, Abell521 (the LoCuSS and Mahdavi+2013 cluster out of the plot in the left panel in Fig. 1), and SPT-CLJ0516-54.

### B.2. Difference of centres for HSE and lensing masses

For some of the clusters, the HSE and lensing masses have been reconstructed assuming different cluster centre positions. We investigate in the following if this miscentring is correlated to the mass ratio and redshift and how it may affect the mass estimates.

#### B.2.1. Correlation between miscentring and bias or redshift

In the left panel in Fig. B.1 we present the ratios between the HSE and lensing masses of the clusters with respect to the separations between the centres considered in the X-ray and lensing analyses. Each marker corresponds to one of the 65 clusters, showing in magenta crosses, purple squares, and grey circles the clusters from ESZ+LoCuSS, LPSZ, and Bartalucci+2018 samples, respectively. Error bars have been obtained from the propagation of the individual XMM-Newton and CoMaLit uncertainties. The separation between X-ray and lensing centres goes from 1.5 kpc to 700 kpc. However, there is no significative correlation between the miscentring and the HSE-to-lensing mass ratio.

The miscentring could also be related to the redshift of the cluster. In the right panel in Fig. B.1 we show the redshift with

respect to the separation of the centres. There is neither indication of correlation. We do not find any significative correlation between the redshift and the uncertainties of mass ratios either.

The dynamical state of clusters is an important point to better understand the evolution of the HSE bias. In particular, relaxed clusters tend to have smaller HSE bias as compared to disturbed ones (Gianfagna et al. 2021). The offset between the centres used in lensing and X-ray analyses could be an indicator of the departure from sphericity and equilibrium of clusters, resulting in a difficulty to define the centre. However, the absence of correlation between the separation of centres and the bias or the redshift does not enable any clear dynamical classification of the clusters in our sample.

#### B.2.2. Simulations to quantify the impact of miscentring

It is evident that two mass estimates computed assuming very different centres are hardly comparable. Nevertheless, setting a quantitative limit of the acceptable separation between the two centres can be difficult. In this section we make use of simulated mock mass density profiles to check the impact that miscentring has on the mass profiles and, consequently, on $M_{500}$.

We simulate mass density profiles using the profiley<fnref n="9" /> Python package, which contains already tested (Madhavacheril et al. 2020) functions that describe density profiles, as well as the corresponding miscentred density profiles. We test a variety of density profile shapes following NFW, gNFW, truncated Navarro-Frenk-White (tNFW, Baltz et al. 2009), and Hernquist (Hernquist 1990) models. To initialise the density profiles we consider different $M_{500}$ (in the range ~ 1 to $9 \times 10^{14}$ M☉) and model parameters. The parameters considered for each model are given in Table B.1.

For each density profile we build the corresponding miscentred profiles by displacing the centre by 0.0 to 0.7 Mpc from the original one. We integrate each miscentred density profile to get a mass profile and obtain the miscentred $M_{500}$ estimate at the radius where the overdensity reaches $\Delta = 500$, $M_{500}^{miscent}$. In Fig. B.2 we show the relative error of the miscentred masses with respect to the separation to the true centre, for all type of profiles and for different redshifts, $z = [0.1, 0.45, 0.8, 1.15, 1.5]$. Each marker indicates a different mass density model and the colours show the true $M_{500}$. For visualisation purposes gNFW, tNFW, and Hernquist markers have been shifted by 0.01, 0.02,

<footnote n="9">⁹ https://profiley.readthedocs.io/en/latest/index.html.</footnote>
<figure><img src="image_16.png" /><figcaption>**Fig. B.1.** Correlation between the HSE-to-lensing mass ratio (left) and redshift (right) of the 65 clusters in the *XMM-Newton*-CoMaLit homogeneous sample with respect to the separation between the centres assumed in the X-ray and lensing analyses. Error bars in the left panel do not account for the systematic scatters.</figcaption></figure>

and 0.03 Mpc. As expected, the figure shows that computing the density profile from a centre that is separated from the true centre gives biased mass reconstructions. This bias increases with the separation, less massive clusters being more sensitive to this effect. The black dashed line in Fig. B.2 indicates $(M_{500}^{\text{miscent}} - M_{500})/M_{500} = 0.5$.

Considering the tested density profiles and mass range, the error due to miscentring is smaller than 50% only if the distance to the real centre is smaller than 100 kpc. Therefore, we decide to exclude clusters for which X-ray and lensing centres are separated by more than 100 kpc. These clusters are: Abell3856, Abell3888, Abell773, Abell665, Abell267, 1E0657-56, Abell521, Abell3376E, Abell520, and Abell2163.

## B.3. Final sample

We present in Table B.2 the 53 clusters in the reference sample. We give their names (as named in the CoMaLit LC² catalogue), redshifts, masses, and mass uncertainties from the analyses of reference, that is, *XMM-Newton* mass reconstruction pipeline masses and CoMaLit estimates.

The histograms in Fig. B.3 show the distributions of the measurement uncertainties for the *XMM-Newton* and CoMaLit masses of all the clusters in the reference sample. The vertical red lines indicate the measured systematic scatter for *XMM-Newton* and CoMaLit masses.

<figure><img src="image_17.png" /><figcaption>**Fig. B.2.** Relative error of $M_{500}$ with respect to the separation between the true and the considered centre for the mass reconstruction. Colours indicate the true $M_{500}$ and markers the mass density model. We consider the same separation for the four models, but gNFW, tNFW, and Hernquist markers have been shifted for visualisation purposes. The black dashed line indicates a 50% error in $M_{500}$.</figcaption></figure>
<figure><img src="image_17.png" /><figcaption>**Fig. B.1.** Correlation between the HSE-to-lensing mass ratio (left) and redshift (right) of the 65 clusters in the *XMM-Newton*-CoMaLit homogeneous sample with respect to the separation between the centres assumed in the X-ray and lensing analyses. Error bars in the left panel do not account for the systematic scatters.</figcaption></figure>

and 0.03 Mpc. As expected, the figure shows that computing the density profile from a centre that is separated from the true centre gives biased mass reconstructions. This bias increases with the separation, less massive clusters being more sensitive to this effect. The black dashed line in Fig. B.2 indicates $(M_{500}^{\text{miscent}} - M_{500})/M_{500} = 0.5$.

Considering the tested density profiles and mass range, the error due to miscentring is smaller than 50% only if the distance to the real centre is smaller than 100 kpc. Therefore, we decide to exclude clusters for which X-ray and lensing centres are separated by more than 100 kpc. These clusters are: Abell3856, Abell3888, Abell773, Abell665, Abell267, 1E0657-56, Abell521, Abell3376E, Abell520, and Abell2163.

### B.3. Final sample

We present in Table B.2 the 53 clusters in the reference sample. We give their names (as named in the CoMaLit LC² catalogue), redshifts, masses, and mass uncertainties from the analyses of reference, that is, *XMM-Newton* mass reconstruction pipeline masses and CoMaLit estimates.

The histograms in Fig. B.3 show the distributions of the measurement uncertainties for the *XMM-Newton* and CoMaLit masses of all the clusters in the reference sample. The vertical red lines indicate the measured systematic scatter for *XMM-Newton* and CoMaLit masses.

<figure><img src="image_18.png" /><figcaption>**Fig. B.2.** Relative error of $M_{500}$ with respect to the separation between the true and the considered centre for the mass reconstruction. Colours indicate the true $M_{500}$ and markers the mass density model. We consider the same separation for the four models, but gNFW, tNFW, and Hernquist markers have been shifted for visualisation purposes. The black dashed line indicates a 50% error in $M_{500}$.</figcaption></figure>
<table><caption>**Table B.1.** Different mass density models and their corresponding parameters used to study the miscentring effect in Fig. B.2.</caption><thead><tr><th>Model</th><th>c<sub>500</sub></th><th>α</th><th>β</th><th>γ</th><th>τ</th><th>η</th></tr></thead><tbody><tr><td>NFW</td><td>1.0, 2.25, 3.5, 4.75, 6.0</td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>Hernquist</td><td>1.0, 2.25, 3.5, 4.75, 6.0</td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>gNFW</td><td>6.0</td><td>0.3, 0.85, 1.4, 1.95, 2.5</td><td>2.0, 3.0, 4.0, 5.0, 6.0</td><td>0.3, 0.475, 0.65, 0.825, 1.0</td><td></td><td></td></tr><tr><td>tNFW</td><td>6.0</td><td></td><td></td><td></td><td>0.2, 1.15, 2.1, 3.05, 4.0</td><td>0.5, 1.125, 1.75, 2.375, 3.0</td></tr></tbody></table>

<figure><img src="image_18.png" /><figcaption>**Fig. B.3.** Comparison between statistical uncertainties and systematic scatter. Dashed line histograms show the statistical uncertainties of all HSE mass estimates from *XMM-Newton* reference analysis (left) and CoMaLit lensing mass estimates (right). Vertical red lines give the systematic scatter estimated with respect to other published results in Sect. 3.3.</figcaption></figure>
<table><caption>**Table B.1.** Different mass density models and their corresponding parameters used to study the miscentring effect in Fig. B.2.</caption><thead><tr><th>Model</th><th>c<sub>500</sub></th><th>α</th><th>β</th><th>γ</th><th>τ</th><th>η</th></tr></thead><tbody><tr><td>NFW</td><td>1.0, 2.25, 3.5, 4.75, 6.0</td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>Hernquist</td><td>1.0, 2.25, 3.5, 4.75, 6.0</td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>gNFW</td><td>6.0</td><td>0.3, 0.85, 1.4, 1.95, 2.5</td><td>2.0, 3.0, 4.0, 5.0, 6.0</td><td>0.3, 0.475, 0.65, 0.825, 1.0</td><td></td><td></td></tr><tr><td>tNFW</td><td>6.0</td><td></td><td></td><td></td><td>0.2, 1.15, 2.1, 3.05, 4.0</td><td>0.5, 1.125, 1.75, 2.375, 3.0</td></tr></tbody></table>

<figure><img src="image_19.png" /><figcaption>**Fig. B.3.** Comparison between statistical uncertainties and systematic scatter. Dashed line histograms show the statistical uncertainties of all HSE mass estimates from *XMM-Newton* reference analysis (left) and CoMaLit lensing mass estimates (right). Vertical red lines give the systematic scatter estimated with respect to other published results in Sect. 3.3.</figcaption></figure>
<table><caption>Table B.2. Characteristics of the reference sample.</caption><thead><tr><th rowspan="2">CoMaLit name</th><th rowspan="2">z</th><th colspan="4">Centre coordinates [deg]</th><th colspan="4">Masses [10<sup>14</sup> M<sub>☉</sub>]</th></tr><tr><th>CoMaLit α</th><th>CoMaLit δ</th><th>XMM α</th><th>XMM δ</th><th>M<sup>CoMaLit lens</sup><sub>500</sub></th><th>δ<sub>M<sup>CoMaLit lens</sup></sub></th><th>M<sup>XMM HSE</sup><sub>500</sub></th><th>δ<sub>M<sup>XMM HSE</sup></sub></th></tr></thead><tbody><tr><td>ABELL2744</td><td>0.3080</td><td>3.57875</td><td>-30.38944</td><td>3.57752</td><td>-30.38633</td><td>13.849</td><td>2.824</td><td>10.472</td><td>2.218</td></tr><tr><td>CL0016+16</td><td>0.5490</td><td>4.63933</td><td>16.43694</td><td>4.63986</td><td>16.43622</td><td>17.821</td><td>4.609</td><td>8.062</td><td>1.147</td></tr><tr><td>ABELL68</td><td>0.2550</td><td>9.27479</td><td>9.16000</td><td>9.27792</td><td>9.15685</td><td>9.171</td><td>1.587</td><td>6.872</td><td>1.181</td></tr><tr><td>ABELL85</td><td>0.0550</td><td>10.46017</td><td>-9.30306</td><td>10.46132</td><td>-9.30438</td><td>5.700</td><td>2.200</td><td>12.150</td><td>2.169</td></tr><tr><td>ABELL2813</td><td>0.2924</td><td>10.85167</td><td>-20.62139</td><td>10.85190</td><td>-20.62294</td><td>8.557</td><td>1.450</td><td>4.131</td><td>0.665</td></tr><tr><td>Abell209</td><td>0.2060</td><td>22.96892</td><td>-13.61122</td><td>22.96912</td><td>-13.61121</td><td>9.614</td><td>1.965</td><td>6.309</td><td>1.010</td></tr><tr><td>ABELL291</td><td>0.1960</td><td>30.43417</td><td>-2.20083</td><td>30.42942</td><td>-2.19678</td><td>4.514</td><td>0.986</td><td>2.718</td><td>0.380</td></tr><tr><td>RXJC0232.2-4420</td><td>0.2840</td><td>38.07750</td><td>-44.34667</td><td>38.07721</td><td>-44.34638</td><td>5.380</td><td>1.815</td><td>6.558</td><td>0.986</td></tr><tr><td>Abell383</td><td>0.1870</td><td>42.01417</td><td>-3.52914</td><td>42.01418</td><td>-3.52889</td><td>5.871</td><td>1.727</td><td>2.629</td><td>0.409</td></tr><tr><td>ABELL478</td><td>0.0881</td><td>63.35667</td><td>10.46694</td><td>63.35581</td><td>10.46371</td><td>8.772</td><td>2.078</td><td>7.043</td><td>0.325</td></tr><tr><td>MS0451.6-0305</td><td>0.5389</td><td>73.54767</td><td>-3.01411</td><td>73.54705</td><td>-3.01620</td><td>9.994</td><td>4.060</td><td>7.352</td><td>1.733</td></tr><tr><td>RXJC0528.9-3927</td><td>0.2840</td><td>82.22083</td><td>-39.47161</td><td>82.22109</td><td>-39.47136</td><td>4.480</td><td>1.310</td><td>9.276</td><td>2.541</td></tr><tr><td>RXJC0532.9-3701</td><td>0.2750</td><td>83.23208</td><td>-37.02667</td><td>83.23226</td><td>-37.02703</td><td>6.960</td><td>1.430</td><td>4.847</td><td>0.852</td></tr><tr><td>SPT-CLJ0546-5345</td><td>1.0660</td><td>86.65321</td><td>-53.76039</td><td>86.65500</td><td>-53.76000</td><td>3.700</td><td>2.760</td><td>4.060</td><td>0.500</td></tr><tr><td>SPT-CLJ0615-5746</td><td>0.9720</td><td>93.96521</td><td>-57.77881</td><td>93.96600</td><td>-57.77960</td><td>4.700</td><td>2.252</td><td>11.960</td><td>1.750</td></tr><tr><td>ABELL3404</td><td>0.1670</td><td>101.37292</td><td>-54.22697</td><td>101.37122</td><td>-54.22732</td><td>8.750</td><td>2.085</td><td>6.360</td><td>0.968</td></tr><tr><td>MACSJ0647.7+7015</td><td>0.5840</td><td>101.95946</td><td>70.24861</td><td>101.95900</td><td>70.24810</td><td>9.427</td><td>2.493</td><td>6.296</td><td>1.014</td></tr><tr><td>MACSJ0911.2+1746</td><td>0.5050</td><td>137.79529</td><td>17.77539</td><td>137.79704</td><td>17.77603</td><td>10.862</td><td>3.259</td><td>3.392</td><td>0.424</td></tr><tr><td>ABELL963</td><td>0.2060</td><td>154.26483</td><td>39.04764</td><td>154.26530</td><td>39.04816</td><td>4.583</td><td>1.637</td><td>4.884</td><td>0.968</td></tr><tr><td>ABELL1300</td><td>0.3080</td><td>172.97583</td><td>-19.92772</td><td>172.97749</td><td>-19.92848</td><td>5.950</td><td>1.695</td><td>5.341</td><td>0.476</td></tr><tr><td>MACSJ1149.5+2223</td><td>0.5440</td><td>177.39871</td><td>22.39850</td><td>177.39763</td><td>22.40108</td><td>14.447</td><td>3.034</td><td>6.536</td><td>1.608</td></tr><tr><td>ABELL1413</td><td>0.1430</td><td>178.82500</td><td>23.40503</td><td>178.82495</td><td>23.40487</td><td>7.200</td><td>2.100</td><td>6.816</td><td>0.815</td></tr><tr><td>MACSJ1206.2-0847</td><td>0.4400</td><td>181.55062</td><td>-8.80094</td><td>181.55208</td><td>-8.80017</td><td>12.176</td><td>2.477</td><td>9.681</td><td>1.203</td></tr><tr><td>ZwCl1215.1+0400</td><td>0.0750</td><td>184.42137</td><td>3.65589</td><td>184.42236</td><td>3.65650</td><td>3.500</td><td>2.200</td><td>5.629</td><td>0.566</td></tr><tr><td>CLJ1226.9+3332</td><td>0.8900</td><td>186.74271</td><td>33.54683</td><td>186.74203</td><td>33.54627</td><td>15.298</td><td>1.275</td><td>4.732</td><td>1.042</td></tr><tr><td>ABELL1576</td><td>0.3010</td><td>189.24583</td><td>63.19056</td><td>189.24408</td><td>63.18711</td><td>13.543</td><td>4.243</td><td>4.064</td><td>0.782</td></tr><tr><td>ABELL1644S</td><td>0.0470</td><td>194.30000</td><td>-17.41306</td><td>194.29601</td><td>-17.41110</td><td>1.309</td><td>0.748</td><td>13.664</td><td>5.775</td></tr><tr><td>ABELL1650</td><td>0.0840</td><td>194.67287</td><td>-1.76139</td><td>194.67336</td><td>-1.76223</td><td>7.100</td><td>2.000</td><td>3.629</td><td>0.315</td></tr><tr><td>ABELL1651</td><td>0.0850</td><td>194.84371</td><td>-4.19603</td><td>194.84380</td><td>-4.19831</td><td>5.600</td><td>2.400</td><td>4.486</td><td>0.429</td></tr><tr><td>ABELL1689</td><td>0.1830</td><td>197.87300</td><td>-1.34100</td><td>197.87263</td><td>-1.34172</td><td>15.033</td><td>1.025</td><td>8.071</td><td>1.078</td></tr><tr><td>ABELL1763</td><td>0.2279</td><td>203.82583</td><td>40.99694</td><td>203.82979</td><td>41.00010</td><td>16.014</td><td>2.050</td><td>5.102</td><td>0.779</td></tr><tr><td>ABELL1795</td><td>0.0620</td><td>207.21871</td><td>26.59300</td><td>207.22115</td><td>26.58994</td><td>9.300</td><td>2.200</td><td>10.910</td><td>0.291</td></tr><tr><td>ABELL1835</td><td>0.2530</td><td>210.25804</td><td>2.87775</td><td>210.25916</td><td>2.87824</td><td>15.510</td><td>4.503</td><td>8.200</td><td>0.660</td></tr><tr><td>PSZ2G099.86+58.45</td><td>0.6160</td><td>213.69662</td><td>54.78433</td><td>213.69522</td><td>54.78396</td><td>7.242</td><td>3.043</td><td>6.421</td><td>2.038</td></tr><tr><td>ABELL1914</td><td>0.1712</td><td>216.50667</td><td>37.82722</td><td>216.51053</td><td>37.82434</td><td>7.929</td><td>1.293</td><td>6.909</td><td>1.390</td></tr><tr><td>ZwCl1454.8+2233</td><td>0.2578</td><td>224.31292</td><td>22.34278</td><td>224.31293</td><td>22.34242</td><td>3.771</td><td>1.457</td><td>3.407</td><td>0.584</td></tr><tr><td>Zwicky7215</td><td>0.2900</td><td>225.34483</td><td>42.34750</td><td>225.34451</td><td>42.34650</td><td>5.390</td><td>1.504</td><td>5.418</td><td>1.142</td></tr><tr><td>ABELL2034</td><td>0.1130</td><td>227.54875</td><td>33.51472</td><td>227.55283</td><td>33.51044</td><td>5.169</td><td>3.100</td><td>5.750</td><td>0.643</td></tr><tr><td>ABELL2029</td><td>0.0770</td><td>227.73371</td><td>5.74481</td><td>227.73502</td><td>5.74410</td><td>12.100</td><td>2.500</td><td>5.592</td><td>0.661</td></tr><tr><td>ABELL2065</td><td>0.0730</td><td>230.62150</td><td>27.70769</td><td>230.62257</td><td>27.70901</td><td>8.000</td><td>2.100</td><td>5.326</td><td>0.474</td></tr><tr><td>ABELL2204</td><td>0.1520</td><td>248.19650</td><td>5.57583</td><td>248.19592</td><td>5.57544</td><td>16.051</td><td>2.963</td><td>7.966</td><td>1.605</td></tr><tr><td>ABELL2218</td><td>0.1760</td><td>248.95329</td><td>66.21417</td><td>248.95972</td><td>66.21254</td><td>8.900</td><td>2.700</td><td>3.829</td><td>0.597</td></tr><tr><td>ABELL2219</td><td>0.2280</td><td>250.08475</td><td>46.70833</td><td>250.08366</td><td>46.71067</td><td>11.729</td><td>1.852</td><td>10.287</td><td>2.969</td></tr><tr><td>RXJ1720.1+2638</td><td>0.1640</td><td>260.04167</td><td>26.62464</td><td>260.04166</td><td>26.62503</td><td>3.510</td><td>1.485</td><td>6.006</td><td>1.314</td></tr><tr><td>Abell2261</td><td>0.2240</td><td>260.61325</td><td>32.13258</td><td>260.61267</td><td>32.13237</td><td>15.613</td><td>3.043</td><td>3.900</td><td>0.590</td></tr><tr><td>MACSJ2129.4-0741</td><td>0.5880</td><td>322.35717</td><td>-7.69189</td><td>322.35913</td><td>-7.69133</td><td>13.486</td><td>3.890</td><td>6.127</td><td>1.179</td></tr><tr><td>RXJ2129.7+0005</td><td>0.2340</td><td>322.41650</td><td>0.08922</td><td>322.41660</td><td>0.08861</td><td>4.470</td><td>1.158</td><td>4.277</td><td>0.491</td></tr><tr><td>ABELL2390</td><td>0.2330</td><td>328.40446</td><td>17.69594</td><td>328.40308</td><td>17.69493</td><td>11.183</td><td>2.396</td><td>9.652</td><td>1.668</td></tr><tr><td>RXJ2228.6+2037</td><td>0.4110</td><td>337.13658</td><td>20.62072</td><td>337.14047</td><td>20.62040</td><td>9.728</td><td>2.626</td><td>6.844</td><td>0.819</td></tr><tr><td>MACSJ2243.3-0935</td><td>0.4470</td><td>340.83933</td><td>-9.59522</td><td>340.83868</td><td>-9.59470</td><td>20.294</td><td>3.865</td><td>8.625</td><td>1.317</td></tr><tr><td>RXJ2248.7-4431</td><td>0.3480</td><td>342.18317</td><td>-44.53092</td><td>342.18243</td><td>-44.53054</td><td>12.400</td><td>3.605</td><td>14.262</td><td>3.762</td></tr><tr><td>ABELL2631</td><td>0.2780</td><td>354.40971</td><td>0.27069</td><td>354.40634</td><td>0.26678</td><td>11.748</td><td>1.888</td><td>9.785</td><td>3.807</td></tr><tr><td>SPT-CLJ2341-5119</td><td>1.0030</td><td>355.30092</td><td>-51.32850</td><td>355.30100</td><td>-51.32860</td><td>1.600</td><td>1.890</td><td>3.190</td><td>0.750</td></tr></tbody></table>

Notes. Column 1: cluster names from the CoMaLit catalogue (entries Comalit_Name and Comalit_Num). Column 2: redshift. Columns 3 to 6: right ascension α and declination δ of the cluster centres according to CoMaLit or X-rays. Columns 7 to 10: cluster masses and uncertainties from the CoMaLit catalogue and from the XMM-Newton analysis.
<table><caption>Table B.2. Characteristics of the reference sample.</caption><thead><tr><th rowspan="2">CoMaLit name</th><th rowspan="2">z</th><th colspan="4">Centre coordinates [deg]</th><th colspan="4">Masses [10<sup>14</sup> M<sub>☉</sub>]</th></tr><tr><th>CoMaLit α</th><th>CoMaLit δ</th><th>XMM α</th><th>XMM δ</th><th>M<sup>CoMaLit lens</sup><sub>500</sub></th><th>δ<sub>M<sup>CoMaLit lens</sup></sub></th><th>M<sup>XMM HSE</sup><sub>500</sub></th><th>δ<sub>M<sup>XMM HSE</sup></sub></th></tr></thead><tbody><tr><td>ABELL2744</td><td>0.3080</td><td>3.57875</td><td>-30.38944</td><td>3.57752</td><td>-30.38633</td><td>13.849</td><td>2.824</td><td>10.472</td><td>2.218</td></tr><tr><td>CL0016+16</td><td>0.5490</td><td>4.63933</td><td>16.43694</td><td>4.63986</td><td>16.43622</td><td>17.821</td><td>4.609</td><td>8.062</td><td>1.147</td></tr><tr><td>ABELL68</td><td>0.2550</td><td>9.27479</td><td>9.16000</td><td>9.27792</td><td>9.15685</td><td>9.171</td><td>1.587</td><td>6.872</td><td>1.181</td></tr><tr><td>ABELL85</td><td>0.0550</td><td>10.46017</td><td>-9.30306</td><td>10.46132</td><td>-9.30438</td><td>5.700</td><td>2.200</td><td>12.150</td><td>2.169</td></tr><tr><td>ABELL2813</td><td>0.2924</td><td>10.85167</td><td>-20.62139</td><td>10.85190</td><td>-20.62294</td><td>8.557</td><td>1.450</td><td>4.131</td><td>0.665</td></tr><tr><td>Abell209</td><td>0.2060</td><td>22.96892</td><td>-13.61122</td><td>22.96912</td><td>-13.61121</td><td>9.614</td><td>1.965</td><td>6.309</td><td>1.010</td></tr><tr><td>ABELL291</td><td>0.1960</td><td>30.43417</td><td>-2.20083</td><td>30.42942</td><td>-2.19678</td><td>4.514</td><td>0.986</td><td>2.718</td><td>0.380</td></tr><tr><td>RXJC0232.2-4420</td><td>0.2840</td><td>38.07750</td><td>-44.34667</td><td>38.07721</td><td>-44.34638</td><td>5.380</td><td>1.815</td><td>6.558</td><td>0.986</td></tr><tr><td>Abell383</td><td>0.1870</td><td>42.01417</td><td>-3.52914</td><td>42.01418</td><td>-3.52889</td><td>5.871</td><td>1.727</td><td>2.629</td><td>0.409</td></tr><tr><td>ABELL478</td><td>0.0881</td><td>63.35667</td><td>10.46694</td><td>63.35581</td><td>10.46371</td><td>8.772</td><td>2.078</td><td>7.043</td><td>0.325</td></tr><tr><td>MS0451.6-0305</td><td>0.5389</td><td>73.54767</td><td>-3.01411</td><td>73.54705</td><td>-3.01620</td><td>9.994</td><td>4.060</td><td>7.352</td><td>1.733</td></tr><tr><td>RXJC0528.9-3927</td><td>0.2840</td><td>82.22083</td><td>-39.47161</td><td>82.22109</td><td>-39.47136</td><td>4.480</td><td>1.310</td><td>9.276</td><td>2.541</td></tr><tr><td>RXJC0532.9-3701</td><td>0.2750</td><td>83.23208</td><td>-37.02667</td><td>83.23226</td><td>-37.02703</td><td>6.960</td><td>1.430</td><td>4.847</td><td>0.852</td></tr><tr><td>SPT-CLJ0546-5345</td><td>1.0660</td><td>86.65321</td><td>-53.76039</td><td>86.65500</td><td>-53.76000</td><td>3.700</td><td>2.760</td><td>4.060</td><td>0.500</td></tr><tr><td>SPT-CLJ0615-5746</td><td>0.9720</td><td>93.96521</td><td>-57.77881</td><td>93.96600</td><td>-57.77960</td><td>4.700</td><td>2.252</td><td>11.960</td><td>1.750</td></tr><tr><td>ABELL3404</td><td>0.1670</td><td>101.37292</td><td>-54.22697</td><td>101.37122</td><td>-54.22732</td><td>8.750</td><td>2.085</td><td>6.360</td><td>0.968</td></tr><tr><td>MACSJ0647.7+7015</td><td>0.5840</td><td>101.95946</td><td>70.24861</td><td>101.95900</td><td>70.24810</td><td>9.427</td><td>2.493</td><td>6.296</td><td>1.014</td></tr><tr><td>MACSJ0911.2+1746</td><td>0.5050</td><td>137.79529</td><td>17.77539</td><td>137.79704</td><td>17.77603</td><td>10.862</td><td>3.259</td><td>3.392</td><td>0.424</td></tr><tr><td>ABELL963</td><td>0.2060</td><td>154.26483</td><td>39.04764</td><td>154.26530</td><td>39.04816</td><td>4.583</td><td>1.637</td><td>4.884</td><td>0.968</td></tr><tr><td>ABELL1300</td><td>0.3080</td><td>172.97583</td><td>-19.92772</td><td>172.97749</td><td>-19.92848</td><td>5.950</td><td>1.695</td><td>5.341</td><td>0.476</td></tr><tr><td>MACSJ1149.5+2223</td><td>0.5440</td><td>177.39871</td><td>22.39850</td><td>177.39763</td><td>22.40108</td><td>14.447</td><td>3.034</td><td>6.536</td><td>1.608</td></tr><tr><td>ABELL1413</td><td>0.1430</td><td>178.82500</td><td>23.40503</td><td>178.82495</td><td>23.40487</td><td>7.200</td><td>2.100</td><td>6.816</td><td>0.815</td></tr><tr><td>MACSJ1206.2-0847</td><td>0.4400</td><td>181.55062</td><td>-8.80094</td><td>181.55208</td><td>-8.80017</td><td>12.176</td><td>2.477</td><td>9.681</td><td>1.203</td></tr><tr><td>ZwCl1215.1+0400</td><td>0.0750</td><td>184.42137</td><td>3.65589</td><td>184.42236</td><td>3.65650</td><td>3.500</td><td>2.200</td><td>5.629</td><td>0.566</td></tr><tr><td>CLJ1226.9+3332</td><td>0.8900</td><td>186.74271</td><td>33.54683</td><td>186.74203</td><td>33.54627</td><td>15.298</td><td>1.275</td><td>4.732</td><td>1.042</td></tr><tr><td>ABELL1576</td><td>0.3010</td><td>189.24583</td><td>63.19056</td><td>189.24408</td><td>63.18711</td><td>13.543</td><td>4.243</td><td>4.064</td><td>0.782</td></tr><tr><td>ABELL1644S</td><td>0.0470</td><td>194.30000</td><td>-17.41306</td><td>194.29601</td><td>-17.41110</td><td>1.309</td><td>0.748</td><td>13.664</td><td>5.775</td></tr><tr><td>ABELL1650</td><td>0.0840</td><td>194.67287</td><td>-1.76139</td><td>194.67336</td><td>-1.76223</td><td>7.100</td><td>2.000</td><td>3.629</td><td>0.315</td></tr><tr><td>ABELL1651</td><td>0.0850</td><td>194.84371</td><td>-4.19603</td><td>194.84380</td><td>-4.19831</td><td>5.600</td><td>2.400</td><td>4.486</td><td>0.429</td></tr><tr><td>ABELL1689</td><td>0.1830</td><td>197.87300</td><td>-1.34100</td><td>197.87263</td><td>-1.34172</td><td>15.033</td><td>1.025</td><td>8.071</td><td>1.078</td></tr><tr><td>ABELL1763</td><td>0.2279</td><td>203.82583</td><td>40.99694</td><td>203.82979</td><td>41.00010</td><td>16.014</td><td>2.050</td><td>5.102</td><td>0.779</td></tr><tr><td>ABELL1795</td><td>0.0620</td><td>207.21871</td><td>26.59300</td><td>207.22115</td><td>26.58994</td><td>9.300</td><td>2.200</td><td>10.910</td><td>0.291</td></tr><tr><td>ABELL1835</td><td>0.2530</td><td>210.25804</td><td>2.87775</td><td>210.25916</td><td>2.87824</td><td>15.510</td><td>4.503</td><td>8.200</td><td>0.660</td></tr><tr><td>PSZ2G099.86+58.45</td><td>0.6160</td><td>213.69662</td><td>54.78433</td><td>213.69522</td><td>54.78396</td><td>7.242</td><td>3.043</td><td>6.421</td><td>2.038</td></tr><tr><td>ABELL1914</td><td>0.1712</td><td>216.50667</td><td>37.82722</td><td>216.51053</td><td>37.82434</td><td>7.929</td><td>1.293</td><td>6.909</td><td>1.390</td></tr><tr><td>ZwCl1454.8+2233</td><td>0.2578</td><td>224.31292</td><td>22.34278</td><td>224.31293</td><td>22.34242</td><td>3.771</td><td>1.457</td><td>3.407</td><td>0.584</td></tr><tr><td>Zwicky7215</td><td>0.2900</td><td>225.34483</td><td>42.34750</td><td>225.34451</td><td>42.34650</td><td>5.390</td><td>1.504</td><td>5.418</td><td>1.142</td></tr><tr><td>ABELL2034</td><td>0.1130</td><td>227.54875</td><td>33.51472</td><td>227.55283</td><td>33.51044</td><td>5.169</td><td>3.100</td><td>5.750</td><td>0.643</td></tr><tr><td>ABELL2029</td><td>0.0770</td><td>227.73371</td><td>5.74481</td><td>227.73502</td><td>5.74410</td><td>12.100</td><td>2.500</td><td>5.592</td><td>0.661</td></tr><tr><td>ABELL2065</td><td>0.0730</td><td>230.62150</td><td>27.70769</td><td>230.62257</td><td>27.70901</td><td>8.000</td><td>2.100</td><td>5.326</td><td>0.474</td></tr><tr><td>ABELL2204</td><td>0.1520</td><td>248.19650</td><td>5.57583</td><td>248.19592</td><td>5.57544</td><td>16.051</td><td>2.963</td><td>7.966</td><td>1.605</td></tr><tr><td>ABELL2218</td><td>0.1760</td><td>248.95329</td><td>66.21417</td><td>248.95972</td><td>66.21254</td><td>8.900</td><td>2.700</td><td>3.829</td><td>0.597</td></tr><tr><td>ABELL2219</td><td>0.2280</td><td>250.08475</td><td>46.70833</td><td>250.08366</td><td>46.71067</td><td>11.729</td><td>1.852</td><td>10.287</td><td>2.969</td></tr><tr><td>RXJ1720.1+2638</td><td>0.1640</td><td>260.04167</td><td>26.62464</td><td>260.04166</td><td>26.62503</td><td>3.510</td><td>1.485</td><td>6.006</td><td>1.314</td></tr><tr><td>Abell2261</td><td>0.2240</td><td>260.61325</td><td>32.13258</td><td>260.61267</td><td>32.13237</td><td>15.613</td><td>3.043</td><td>3.900</td><td>0.590</td></tr><tr><td>MACSJ2129.4-0741</td><td>0.5880</td><td>322.35717</td><td>-7.69189</td><td>322.35913</td><td>-7.69133</td><td>13.486</td><td>3.890</td><td>6.127</td><td>1.179</td></tr><tr><td>RXJ2129.7+0005</td><td>0.2340</td><td>322.41650</td><td>0.08922</td><td>322.41660</td><td>0.08861</td><td>4.470</td><td>1.158</td><td>4.277</td><td>0.491</td></tr><tr><td>ABELL2390</td><td>0.2330</td><td>328.40446</td><td>17.69594</td><td>328.40308</td><td>17.69493</td><td>11.183</td><td>2.396</td><td>9.652</td><td>1.668</td></tr><tr><td>RXJ2228.6+2037</td><td>0.4110</td><td>337.13658</td><td>20.62072</td><td>337.14047</td><td>20.62040</td><td>9.728</td><td>2.626</td><td>6.844</td><td>0.819</td></tr><tr><td>MACSJ2243.3-0935</td><td>0.4470</td><td>340.83933</td><td>-9.59522</td><td>340.83868</td><td>-9.59470</td><td>20.294</td><td>3.865</td><td>8.625</td><td>1.317</td></tr><tr><td>RXJ2248.7-4431</td><td>0.3480</td><td>342.18317</td><td>-44.53092</td><td>342.18243</td><td>-44.53054</td><td>12.400</td><td>3.605</td><td>14.262</td><td>3.762</td></tr><tr><td>ABELL2631</td><td>0.2780</td><td>354.40971</td><td>0.27069</td><td>354.40634</td><td>0.26678</td><td>11.748</td><td>1.888</td><td>9.785</td><td>3.807</td></tr><tr><td>SPT-CLJ2341-5119</td><td>1.0030</td><td>355.30092</td><td>-51.32850</td><td>355.30100</td><td>-51.32860</td><td>1.600</td><td>1.890</td><td>3.190</td><td>0.750</td></tr></tbody></table>

Notes. Column 1: cluster names from the CoMaLit catalogue (entries Comalit_Name and Comalit_Num). Column 2: redshift. Columns 3 to 6: right ascension α and declination δ of the cluster centres according to CoMaLit or X-rays. Columns 7 to 10: cluster masses and uncertainties from the CoMaLit catalogue and from the XMM-Newton analysis.
# Appendix C: Different scaling relation models

## C.1. Additional figures

In this section we present additional figures of the SR model extension fits described in Sect. 5.2. We show in Fig. C.1 and C.2 the scaling relations and posterior parameter distributions for different subsamples defined according to the redshift range, in these cases not including the systematic scatters in the error bars of each cluster. In Fig. C.1 we present the scaling relations for the samples with $z < 0.2$ and $z > 0.2$, while the panels in Fig. C.2 show the results for a $z < 0.5$ sample.

Figures C.3 and C.4 compare the SR and parameters posterior distributions when considering and not redshift evolution.

## C.2. Comparison of SR models with AIC and BIC

In Fig. C.5 we present the $\hat{\chi}^2$ (defined in Eq. 15) distributions normalised by the degrees of freedom (d.o.f.) in each SR model fit as solid line histograms. In the left panel we present the results for the scaling relations fitted in $\ln M^{\text{HSE}} - \ln M^{\text{lens}}$ and in the right for $M^{\text{HSE}} - M^{\text{lens}}$. The blue histogram in the left panel shows the reduced $\hat{\chi}^2$ for the scaling relation of reference in this work and the vertical solid line is the median value of the distribution. For comparison, the dashed lines correspond to the $\chi^2$-distributions,

$$f(\chi^2) = \frac{1}{2^{\nu/2}\Gamma(\nu/2)} e^{-\chi^2/2}(\chi^2)^{(\nu/2)-1}, \qquad (C.1)$$

<figure><img src="image_19.png" /><figcaption>Fig. C.1. Same as Fig. 6, but without considering the systematic scatter in the fit.</figcaption></figure>

where $\nu$ is the number of the degrees of freedom (d.o.f.). For the scaling relation of reference we have d.o.f. = 53 - 3 = 50. The red and green results show the reduced $\hat{\chi}^2$ for the fits of the scaling relations when considering a deviation from linearity and an evolution with redshift, respectively. The histograms follow fairly well the $\chi^2$-distributions with $\nu$ degrees of freedom.

We use the Akaike information criterion (AIC, Akaike 1974) and the Bayesian information criterion (BIC, Schwarz 1978) to compare the improvement in the $\hat{\chi}^2$ when adding parameters to the model. We calculate:

$$\text{AIC} = \hat{\chi}_{\text{min}}^2 + 2K, \qquad (C.2)$$

and

$$\text{BIC} = \hat{\chi}_{\text{min}}^2 + K \ln N, \qquad (C.3)$$

where $\hat{\chi}_{\text{min}}^2$ is the minimum of the $\hat{\chi}^2$ values for each model. Here $K$ and $N$ are the number of free parameters in the model and the total number of data points, that is, $N = N_{\text{clusters}} = 53$, respectively.

We report in Table C.1 the results for the different scaling relation models and the $\Delta$AIC and $\Delta$BIC differences with respect to the simplest scaling law amongst the nested models.

<figure><img src="image_20.png" /></figure>
# Appendix C: Different scaling relation models

## C.1. Additional figures

In this section we present additional figures of the SR model extension fits described in Sect. 5.2. We show in Fig. C.1 and C.2 the scaling relations and posterior parameter distributions for different subsamples defined according to the redshift range, in these cases not including the systematic scatters in the error bars of each cluster. In Fig. C.1 we present the scaling relations for the samples with $z < 0.2$ and $z > 0.2$, while the panels in Fig. C.2 show the results for a $z < 0.5$ sample.

Figures C.3 and C.4 compare the SR and parameters posterior distributions when considering and not redshift evolution.

## C.2. Comparison of SR models with AIC and BIC

In Fig. C.5 we present the $\hat{\chi}^2$ (defined in Eq. 15) distributions normalised by the degrees of freedom (d.o.f.) in each SR model fit as solid line histograms. In the left panel we present the results for the scaling relations fitted in $\ln M^{\text{HSE}} - \ln M^{\text{lens}}$ and in the right for $M^{\text{HSE}} - M^{\text{lens}}$. The blue histogram in the left panel shows the reduced $\hat{\chi}^2$ for the scaling relation of reference in this work and the vertical solid line is the median value of the distribution. For comparison, the dashed lines correspond to the $\chi^2$-distributions,

$$f(\chi^2) = \frac{1}{2^{\nu/2}\Gamma(\nu/2)} e^{-\chi^2/2}(\chi^2)^{(\nu/2)-1}, \quad (C.1)$$

<figure><img src="image_20.png" /><figcaption>Fig. C.1. Same as Fig. 6, but without considering the systematic scatter in the fit.</figcaption></figure>

where $\nu$ is the number of the degrees of freedom (d.o.f.). For the scaling relation of reference we have d.o.f. = 53 - 3 = 50. The red and green results show the reduced $\hat{\chi}^2$ for the fits of the scaling relations when considering a deviation from linearity and an evolution with redshift, respectively. The histograms follow fairly well the $\chi^2$-distributions with $\nu$ degrees of freedom.

We use the Akaike information criterion (AIC, Akaike 1974) and the Bayesian information criterion (BIC, Schwarz 1978) to compare the improvement in the $\hat{\chi}^2$ when adding parameters to the model. We calculate:

$$\text{AIC} = \hat{\chi}_{\text{min}}^2 + 2K, \quad (C.2)$$

and

$$\text{BIC} = \hat{\chi}_{\text{min}}^2 + K \ln N, \quad (C.3)$$

where $\hat{\chi}_{\text{min}}^2$ is the minimum of the $\hat{\chi}^2$ values for each model. Here $K$ and $N$ are the number of free parameters in the model and the total number of data points, that is, $N = N_{\text{clusters}} = 53$, respectively.

We report in Table C.1 the results for the different scaling relation models and the $\Delta$AIC and $\Delta$BIC differences with respect to the simplest scaling law amongst the nested models.

<figure><img src="image_21.png" /></figure>
<figure><img src="image_21.png" /><figcaption>**Fig. C.2.** Scaling relation between HSE and lensing masses for the reference sample in grey and a subsample containing only *z* < 0.5 clusters in cyan, without considering the systematic scatter in the fit. The black dashed line shows the equality. The corner plots in the right panel are the posterior 1D and 2D distributions of the parameters in the SR. Here *β*<sup>HSE</sup> is fixed to 1.</figcaption></figure>

<figure><img src="image_22.png" /><figcaption>**Fig. C.3.** Same as Fig. 5 but with dashed lines showing the results if an evolution with redshift is considered in the scaling relation and solid lines without evolution.</figcaption></figure>

<table><caption>**Table C.1.** Statistical results for the scaling relation models presented in this work.</caption><thead><tr><th>Model</th><th><i>K</i></th><th><i>N</i></th><th><i>χ</i><sup>2</sup><sub>min</sub></th><th>AIC</th><th>ΔAIC</th><th>BIC</th><th>ΔBIC</th></tr></thead><tbody><tr><td>Reference SR</td><td>3</td><td>53</td><td>18.126</td><td>24.126</td><td>0.000</td><td>21.422</td><td>0.000</td></tr><tr><td>Deviation from linearity</td><td>4</td><td>53</td><td>16.020</td><td>24.020</td><td>-0.106</td><td>21.565</td><td>+0.143</td></tr><tr><td>Evolution with redshift</td><td>4</td><td>53</td><td>22.784</td><td>30.784</td><td>+6.658</td><td>28.329</td><td>+6.907</td></tr><tr><td>Considering an offset</td><td>4</td><td>53</td><td>13.929</td><td>21.993</td><td>0.000</td><td>19.475</td><td>0.000</td></tr><tr><td>Evolution with redshift and offset</td><td>5</td><td>53</td><td>17.992</td><td>27.992</td><td>+5.999</td><td>26.039</td><td>+6.564</td></tr></tbody></table>

**Notes.** We report the values for the fits accounting for the systematic scatter of HSE and lensing masses.
<figure><img src="image_22.png" /><figcaption>**Fig. C.2.** Scaling relation between HSE and lensing masses for the reference sample in grey and a subsample containing only *z* < 0.5 clusters in cyan, without considering the systematic scatter in the fit. The black dashed line shows the equality. The corner plots in the right panel are the posterior 1D and 2D distributions of the parameters in the SR. Here *β*<sup>HSE</sup> is fixed to 1.</figcaption></figure>

<figure><img src="image_23.png" /><figcaption>**Fig. C.3.** Same as Fig. 5 but with dashed lines showing the results if an evolution with redshift is considered in the scaling relation and solid lines without evolution.</figcaption></figure>

<table><caption>**Table C.1.** Statistical results for the scaling relation models presented in this work.</caption><thead><tr><th>Model</th><th><i>K</i></th><th><i>N</i></th><th><i>x̂</i><sup>2</sup><sub>min</sub></th><th>AIC</th><th>ΔAIC</th><th>BIC</th><th>ΔBIC</th></tr></thead><tbody><tr><td>Reference SR</td><td>3</td><td>53</td><td>18.126</td><td>24.126</td><td>0.000</td><td>21.422</td><td>0.000</td></tr><tr><td>Deviation from linearity</td><td>4</td><td>53</td><td>16.020</td><td>24.020</td><td>-0.106</td><td>21.565</td><td>+0.143</td></tr><tr><td>Evolution with redshift</td><td>4</td><td>53</td><td>22.784</td><td>30.784</td><td>+6.658</td><td>28.329</td><td>+6.907</td></tr><tr><td>Considering an offset</td><td>4</td><td>53</td><td>13.929</td><td>21.993</td><td>0.000</td><td>19.475</td><td>0.000</td></tr><tr><td>Evolution with redshift and offset</td><td>5</td><td>53</td><td>17.992</td><td>27.992</td><td>+5.999</td><td>26.039</td><td>+6.564</td></tr></tbody></table>

**Notes.** We report the values for the fits accounting for the systematic scatter of HSE and lensing masses.
<figure><img src="image_23.png" /><figcaption>**Fig. C.4.** Same as Fig. 9 but with dashed lines showing the results if an evolution with redshift is considered in the scaling relation and solid lines without evolution.</figcaption></figure>

<figure><img src="image_24.png" /><figcaption>**Fig. C.5.** Comparison of the reduced $\hat{\chi}^2$ distributions for different scaling relation fits (solid colour histograms) and the $\chi^2$-distribution (Eq. C.1) for the degrees of freedom in each fit (dashed colour lines). Vertical lines show the median value of each histogram.</figcaption></figure>
<figure><img src="image_24.png" /><figcaption>**Fig. C.4.** Same as Fig. 9 but with dashed lines showing the results if an evolution with redshift is considered in the scaling relation and solid lines without evolution.</figcaption></figure>

<figure><img src="image_25.png" /><figcaption>**Fig. C.5.** Comparison of the reduced $\hat{\chi}^2$ distributions for different scaling relation fits (solid colour histograms) and the $\chi^2$-distribution (Eq. C.1) for the degrees of freedom in each fit (dashed colour lines). Vertical lines show the median value of each histogram.</figcaption></figure>
## Appendix D: Other HSE mass estimates from XMM-Newton data

As described in Sect. 2.1.2, all the XMM-Newton HSE masses used in this work were obtained from mass profiles, by interpolating the binned profile to get $M_{500}$ and $R_{500}$. For all the clusters in our reference sample we also have access to X-ray masses obtained with the $Y_X - M_{500}^{HSE}$ scaling relation from Arnaud et al. (2010). The quantity $Y_X$ is defined as the product of the gas mass ($M_{g,500}$) and the spectroscopic temperature ($T_X$) and it is the X-ray analogue of the integrated Compton parameter (Kravtsov et al. 2006). We use these masses as $M_{500}^{HSE}$ estimates and fit again the scaling relations in Eq. 9, 10, 11, and 12.

Median values and uncertainties of the fitted SR parameters are given in Table D.1. The intrinsic scatters of HSE and lensing masses with respect to the SR are reduced when using

$Y_X - M_{500}^{HSE}$ estimates and the measured HSE bias is also smaller (larger $1-b$).

For ESZ+LoCuSS and LPSZ sample clusters we have a third estimate of the HSE mass, obtained by fitting an NFW mass model to the X-ray profiles. Such masses tend to be in agreement with the interpolated ones (the reference XMM-Newton masses used in this work, Sect. 2.1.2), with a mean and median ratio for the 50 cluster estimates of $M_{500}/M_{500}^{NFW} \sim 1.11$ and 0.95, indicating that the HSE masses used in the analyses of reference are robust against modelling effects. Masses obtained by fitting an NFW model have in median 30% smaller uncertainties than the interpolated estimates. The scaling relations obtained with NFW fit masses (Table D.1) and interpolated ones (second row in Table 3 for the 50 clusters with $z < 0.9$) are almost identical, with $\alpha^{HSE}$ centred in the exact same value, but with $10-20\%$ smaller intrinsic scatters when using NFW masses.

<table><caption>**Table D.1.** Summary of the median values and uncertainties at the 16th and 84th percentiles of the fitted parameters for the HSE-to-lensing mass SR for HSE masses obtained from the $Y_X - M_{500}^{HSE}$ scaling relation and from the fit of an NFW model to the X-ray data.</caption><thead><tr><th>Sample</th><th># of clusters</th><th colspan="5">No σ<sup>2</sup><sub>sys</sub></th></tr><tr><th>Y<sub>X</sub> - M<sub>500</sub><sup>HSE</sup></th><th></th><th>α<sup>HSE</sup></th><th>e<sup>α<sup>HSE</sup></sup> = (1 - b)</th><th>β<sup>HSE</sup></th><th>σ<sup>HSE</sup></th><th>σ<sup>lens</sup></th></tr></thead><tbody><tr><td>Reference sample</td><td>53</td><td>-0.237<sup>+0.095</sup><sub>-0.087</sub></td><td>0.789<sup>+0.075</sup><sub>-0.069</sub></td><td>[1]</td><td>0.105<sup>+0.063</sup><sub>-0.067</sub></td><td>0.247<sup>+0.058</sup><sub>-0.056</sub></td></tr><tr><td>Reference sample</td><td>53</td><td>-0.131<sup>+0.542</sup><sub>-0.631</sub></td><td></td><td>0.953<sup>+0.296</sup><sub>-0.263</sub></td><td>0.133<sup>+0.075</sup><sub>-0.088</sub></td><td>0.231<sup>+0.068</sup><sub>-0.098</sub></td></tr><tr><td>Reference sample (BCES)</td><td>53</td><td>0.504 ± 0.262</td><td></td><td>0.662 ± 0.122</td><td>0.250*</td><td>-</td></tr><tr><td></td><td></td><th>A<sup>HSE</sup> [10<sup>14</sup> M<sub>☉</sub>]</th><th>B<sup>HSE</sup></th><th colspan="3">σ<sup>HSE</sup> [10<sup>14</sup> M<sub>☉</sub>] σ<sup>lens</sup> [10<sup>14</sup> M<sub>☉</sub>]</th></tr><tr><td>Reference sample</td><td>53</td><td>0.990<sup>+1.748</sup><sub>-2.050</sub></td><td>0.672<sup>+0.235</sup><sub>-0.205</sub></td><td colspan="3"></td></tr><tr><td>Reference sample (BCES)</td><td>53</td><td>2.401 ± 0.781</td><td>0.487 ± 0.095</td><td colspan="3">1.480*</td></tr><tr><th>NFW fit</th><th></th><th>α<sup>HSE</sup></th><th>e<sup>α<sup>HSE</sup></sup> = (1 - b)</th><th>β<sup>HSE</sup></th><th>σ<sup>HSE</sup></th><th>σ<sup>lens</sup></th></tr><tr><td>z < 0.9</td><td>50</td><td>-0.309<sup>+0.119</sup><sub>-0.107</sub></td><td>0.734<sup>+0.087</sup><sub>-0.079</sub></td><td>[1]</td><td>0.249<sup>+0.059</sup><sub>-0.063</sub></td><td>0.212<sup>+0.077</sup><sub>-0.086</sub></td></tr><tr><td>z < 0.9</td><td>50</td><td>-0.361<sup>+0.692</sup><sub>-0.955</sub></td><td></td><td>1.030<sup>+0.480</sup><sub>-0.341</sub></td><td>0.246<sup>+0.073</sup><sub>-0.121</sub></td><td>0.220<sup>+0.081</sup><sub>-0.119</sub></td></tr><tr><td>z < 0.9 (BCES)</td><td>50</td><td>0.032 ± 0.485</td><td></td><td>0.836 ± 0.222</td><td>0.329*</td><td>-</td></tr><tr><td></td><td></td><th>A<sup>HSE</sup> [10<sup>14</sup> M<sub>☉</sub>]</th><th>B<sup>HSE</sup></th><th colspan="3">σ<sup>HSE</sup> [10<sup>14</sup> M<sub>☉</sub>] σ<sup>lens</sup> [10<sup>14</sup> M<sub>☉</sub>]</th></tr><tr><td>z < 0.9</td><td>50</td><td>1.205<sup>+1.490</sup><sub>-2.463</sub></td><td>0.579<sup>+0.287</sup><sub>-0.173</sub></td><td colspan="3"></td></tr><tr><td>z < 0.9 (BCES)</td><td>50</td><td>0.530 ± 1.343</td><td>0.646 ± 0.159</td><td colspan="3">1.944*</td></tr></tbody></table>

**Notes.** We present the results assuming linearity, a deviation from linearity, and an offset between HSE and lensing masses. For the BCES fit, we report the best-fit values and $1\sigma$ uncertainties. (*) We also calculate the scatter with respect to the best BCES scaling relations following Eq. 4.
## Appendix D: Other HSE mass estimates from XMM-Newton data

As described in Sect. 2.1.2, all the XMM-Newton HSE masses used in this work were obtained from mass profiles, by interpolating the binned profile to get $M_{500}$ and $R_{500}$. For all the clusters in our reference sample we also have access to X-ray masses obtained with the $Y_X - M_{500}^{HSE}$ scaling relation from Arnaud et al. (2010). The quantity $Y_X$ is defined as the product of the gas mass ($M_{g,500}$) and the spectroscopic temperature ($T_X$) and it is the X-ray analogue of the integrated Compton parameter (Kravtsov et al. 2006). We use these masses as $M_{500}^{HSE}$ estimates and fit again the scaling relations in Eq. 9, 10, 11, and 12.

Median values and uncertainties of the fitted SR parameters are given in Table D.1. The intrinsic scatters of HSE and lensing masses with respect to the SR are reduced when using

$Y_X - M_{500}^{HSE}$ estimates and the measured HSE bias is also smaller (larger $1-b$).

For ESZ+LoCuSS and LPSZ sample clusters we have a third estimate of the HSE mass, obtained by fitting an NFW mass model to the X-ray profiles. Such masses tend to be in agreement with the interpolated ones (the reference XMM-Newton masses used in this work, Sect. 2.1.2), with a mean and median ratio for the 50 cluster estimates of $M_{500}/M_{500}^{NFW} \sim 1.11$ and 0.95, indicating that the HSE masses used in the analyses of reference are robust against modelling effects. Masses obtained by fitting an NFW model have in median 30% smaller uncertainties than the interpolated estimates. The scaling relations obtained with NFW fit masses (Table D.1) and interpolated ones (second row in Table 3 for the 50 clusters with $z < 0.9$) are almost identical, with $\alpha^{HSE}$ centred in the exact same value, but with 10-20% smaller intrinsic scatters when using NFW masses.

<table><caption>**Table D.1.** Summary of the median values and uncertainties at the 16th and 84th percentiles of the fitted parameters for the HSE-to-lensing mass SR for HSE masses obtained from the $Y_X - M_{500}^{HSE}$ scaling relation and from the fit of an NFW model to the X-ray data.</caption><thead><tr><th>Sample</th><th># of clusters</th><th colspan="5">No σ<sup>2</sup><sub>sys</sub></th></tr><tr><th>Y<sub>X</sub> - M<sub>500</sub><sup>HSE</sup></th><th></th><th>α<sup>HSE</sup></th><th>e<sup>α<sup>HSE</sup></sup> = (1 - b)</th><th>β<sup>HSE</sup></th><th>σ<sup>HSE</sup></th><th>σ<sup>lens</sup></th></tr></thead><tbody><tr><td>Reference sample</td><td>53</td><td>-0.237<sup>+0.095</sup><sub>-0.087</sub></td><td>0.789<sup>+0.075</sup><sub>-0.069</sub></td><td>[1]</td><td>0.105<sup>+0.063</sup><sub>-0.067</sub></td><td>0.247<sup>+0.058</sup><sub>-0.056</sub></td></tr><tr><td>Reference sample</td><td>53</td><td>-0.131<sup>+0.542</sup><sub>-0.631</sub></td><td></td><td>0.953<sup>+0.296</sup><sub>-0.263</sub></td><td>0.133<sup>+0.075</sup><sub>-0.088</sub></td><td>0.231<sup>+0.068</sup><sub>-0.098</sub></td></tr><tr><td>Reference sample (BCES)</td><td>53</td><td>0.504 ± 0.262</td><td></td><td>0.662 ± 0.122</td><td>0.250*</td><td>-</td></tr><tr><td></td><td></td><th>A<sup>HSE</sup> [10<sup>14</sup> M<sub>☉</sub>]</th><th>B<sup>HSE</sup></th><th colspan="3">σ<sup>HSE</sup> [10<sup>14</sup> M<sub>☉</sub>]</th><th>σ<sup>lens</sup> [10<sup>14</sup> M<sub>☉</sub>]</th></tr><tr><td>Reference sample</td><td>53</td><td>0.990<sup>+1.748</sup><sub>-2.050</sub></td><td>0.672<sup>+0.235</sup><sub>-0.205</sub></td><td colspan="3">0.930<sup>+0.508</sup><sub>-0.633</sub></td><td>2.333<sup>+0.594</sup><sub>-1.032</sub></td></tr><tr><td>Reference sample (BCES)</td><td>53</td><td>2.401 ± 0.781</td><td>0.487 ± 0.095</td><td colspan="3">1.480*</td><td>-</td></tr><tr><th>NFW fit</th><th></th><th>α<sup>HSE</sup></th><th>e<sup>α<sup>HSE</sup></sup> = (1 - b)</th><th>β<sup>HSE</sup></th><th>σ<sup>HSE</sup></th><th>σ<sup>lens</sup></th></tr><tr><td>z < 0.9</td><td>50</td><td>-0.309<sup>+0.119</sup><sub>-0.107</sub></td><td>0.734<sup>+0.087</sup><sub>-0.079</sub></td><td>[1]</td><td>0.249<sup>+0.059</sup><sub>-0.063</sub></td><td>0.212<sup>+0.077</sup><sub>-0.086</sub></td></tr><tr><td>z < 0.9</td><td>50</td><td>-0.361<sup>+0.692</sup><sub>-0.955</sub></td><td></td><td>1.030<sup>+0.480</sup><sub>-0.341</sub></td><td>0.246<sup>+0.073</sup><sub>-0.121</sub></td><td>0.220<sup>+0.081</sup><sub>-0.119</sub></td></tr><tr><td>z < 0.9 (BCES)</td><td>50</td><td>0.032 ± 0.485</td><td></td><td>0.836 ± 0.222</td><td>0.329*</td><td>-</td></tr><tr><td></td><td></td><th>A<sup>HSE</sup> [10<sup>14</sup> M<sub>☉</sub>]</th><th>B<sup>HSE</sup></th><th colspan="3">σ<sup>HSE</sup> [10<sup>14</sup> M<sub>☉</sub>]</th><th>σ<sup>lens</sup> [10<sup>14</sup> M<sub>☉</sub>]</th></tr><tr><td>z < 0.9</td><td>50</td><td>1.205<sup>+1.490</sup><sub>-2.463</sub></td><td>0.579<sup>+0.287</sup><sub>-0.173</sub></td><td colspan="3">1.576<sup>+0.408</sup><sub>-0.589</sub></td><td>1.989<sup>+0.843</sup><sub>-1.164</sub></td></tr><tr><td>z < 0.9 (BCES)</td><td>50</td><td>0.530 ± 1.343</td><td>0.646 ± 0.159</td><td colspan="3">1.944*</td><td>-</td></tr></tbody></table>

**Notes.** We present the results assuming linearity, a deviation from linearity, and an offset between HSE and lensing masses. For the BCES fit, we report the best-fit values and $1\sigma$ uncertainties. (*) We also calculate the scatter with respect to the best BCES scaling relations following Eq. 4.