## 2 Theory  
### Nuclear-Model Calculations  
In the present work, we adopt the nuclear shell model (NSM) and the proton-neutron quasiparticle random-phase approximation (pnQRPA) [34] as the basic nuclear-modelframeworks. We compute the wave functions of the states of the odd-odd nucleus \({}^{136}\)Cs by using these models in order to access the \(0\
u\beta\beta\)-decay amplitudes in a non-closure approach and compare them with the corresponding OMC rates computed in Reference [33]. In both the OMC and \(0\
u\beta\beta\) calculations we use the phenomenological NSM (**sm-phen**) and pnQRPA (**qrpa-phen**) approach, with the relevant parameters defined in Table 1 of Reference [33]. As in [33], the decay amplitudes are computed for states with excitation energy \(\leq 1\) MeV, an energy range relevant to present-day MONUMENT experiment [31].  
In order to give the reader a brief glimpse of the parameters used in the calculations, we repeat here some of the information given in full length in Ref. [33]. For the NSM calculations, we chose the jj55 pn model space with the \(2s\)-\(1d\)-\(0g_{7/2}\)-\(0h_{11/2}\) set of single-particle orbitals for both protons and neutrons. We use the sn100 pn interaction [35] and we use the quenching factor \(q=0.74\) benchmarked by the works [36; 37; 38], leading to the effective axial coupling of \(g_{A}^{eff}=0.93\). We determine the value of the pseudoscalar coupling \(g_{P}\) by using the Godberger-Treiman partially conserved axial vector current (PCAC) hypothesis \(g_{P}/g_{A}\approx 6.8\). For more details, see [33].  
For the pnQRPA, we use the no-core valence space of Ref. [39] based on the Woods-Saxon parameters of Ref. [40] modified slightly at the proton and neutron Fermi surfaces in order to better reproduce the single-quasiparticle type of spectra of the neighboring odd-\(A\) nuclei. We use the Bonn-A G-matrix interaction [41] with the BCS (Bardeen-Cooper-Schriefer) mean field defined by the pairing-parameter values \(g_{p}^{pair}=0.83\) for protons and \(g_{n}^{pair}=0.87\) for neutrons, fitted to reproduce the available phenomenological proton and neutron separation energies in \({}^{136}\)Ba [34]. The beyond-BCS mean field effective residual interaction is defined using the particle-hole parameter \(g_{ph}=1.18\), fitted to the experimental energy of the Gamow-Teller giant resonance. For the particle-particle channel, we use the renormalization scheme introduced in [42] by dividing the corresponding strength parameter to the isoscalar \(g_{pp}^{T=0}=0.7\) and isovector \(g_{pp}^{T=1}=0.7\) parts, the values taken from Ref. [39]. For the weak axial coupling, we use the value \(g_{A}^{eff}=0.83\) taken from the systematics obtained in [43]. For the weak pseudoscalar coupling, we use the recipe adopted for the NSM above.  
An interesting additional point could be raised here: Our adopted pnQRPA framework is based on a spherical single-particle mean field. This should be contrasted with the fact that the nucleus \({}^{136}\)Xe is spherical but the nucleus \({}^{136}\)Ba shows signs of deformation, as seen, e.g., in its excitation spectrum. This mismatch of shapes of the \(0\
u\beta\beta\) mother and daughter nuclei is a well-known driver of the suppression of the magnitude \(0\
u\beta\beta\) NME through the pnQRPA overlap factor present in Equation (8) below [44; 45; 46]. This feature naturally affects the accuracy of both the OMC rate calculations and the right-leg virtual amplitudes of the \(0\
u\beta\beta\) decay. However, as both the OMC and \(0\
u\beta\beta\) calculations are affected by the same inaccuracy, the correlations found between the two processes are most likely not altered much. A similar argument supports the view that also the correlations found in the NSM calculations are robust and are not affected by the difference in deformations of the two nuclei.