## 2 Materials and Methods  
Synthesis, characterization, and electrode preparation. Sodium-rich manganese hexacyanoferrate (MnHCF) was synthesized, characterized, and processed for electrode use, as previously reported [11]. The chemical formula of the synthesized material can be expressed as Na\({}_{1.9}\)Mn\({}_{1.1}\)Fe (CN)\({}_{6}\). The electrode slurry consisted of the active material (MnHCF, 85 wt%), the conductive agent (Super C65, IMERYS, 10 wt%), and the binder (polyvinylidene dilluoride, PVDF; Solef 6020, Solvay Polymerspecialties, 5 wt%). The areal loading of the electrodes had a Student's \(t\)-distribution of 2.47 \(\pm\) 0.04 mg\({}_{\text{MnHCF}}\) cm\({}^{-2}\), considering a population of 24 electrodes, and a confidence interval equal to 0.95. The final electrodes were extensively dried at 120 \({}^{\circ}\)C under vacuum (10\({}^{-6}\) bar) for 24 h and finally stored in an Ar-filled glove box. For the detailed description of the synthesis, the physicochemical characterization, and the electrode processing, please refer to reference [11]. Operando XAFS measurements and data analysis. A suitable electrochemical cell for in situ experiments, described in detail elsewhere [14], was used for the XAFS measurements. A large piece of lithium metal foil was used as the negative electrode, while the MnHCF-based electrode was used as the positive electrode. LP30, i.e., 1.0 M LiPF\({}_{6}\) dissolved in a 1:1 volumetric mixture of ethylene carbonate and dimethyl carbonate (EC/DMC = 50/50 _v/v_), was adopted as the electrolyte solution. The positive electrode, glass fiber separator, and negative electrode were stacked and assembled under inert atmosphere in an Ar-filled glovebox. Galvanostatic cycling with potential limitation (GCPL) was conducted by considering 1C current rate equal to 150 mAhg\({}^{-1}\). The operando experiment was performed during one galvanostatic cycle at C/15 current rate (corresponding to a current density equal to 0.01 A g\({}_{\text{MnHCF}}\)\({}^{-1}\)), starting with the charge and by recording a full set of spectra from number 4 to number 123. Before the operando experiment, the cell was subjected to an anomalous cycle test, consisting of five cycles with current density set at \(\pm\)10 A g\({}_{\text{MnHCF}}\)\({}^{-1}\) (i.e., 67 C), to simulate BMS malfunctioning and ill-usage of the device, followed by a discharge with current density equal to \(-\)0.1 A g\({}_{\text{MnHCF}}\)\({}^{-1}\) (see Figure S1) [11; 12]. XAFS spectra were recorded at the ROCK (Rocking Optics for Chemical Kinetics) beamline of the SOLEIL synchrotron [15]. Spectra were alternatingly collected from 6380 eV to 7080 eV and from 7000 eV to 7969 eV, i.e., around the Mn and Fe K-edges, respectively. The energies were calibrated by assigning the first inflection points of the spectra of manganese and of iron metal to 6539 and 7112 eV, respectively. XAFS data pre-treatment was carried out using ATHENA graphical utility [16]. The normalized XAFS spectra were further analyzed using a chemometric approach, implying a combination of principal component analysis (PCA) with Multivariate Curve Resolution--Alternating Least Square (MCR-ALS) using a specifically developed toolbox in MATLAB [17] to retrieve mathematically the pure spectral components and the concentration profile plots from the experimental spectroscopic matrix. This analytical approach is described in detail in reference [18]. The extended X-ray absorption fine structure (EXAFS) analysis was performed using the GNXAS package [19; 20], which is based on the decomposition of the EXAFS signals into a sum of two-body \(\gamma^{(2)}\), three-body \(\gamma^{(3)}\), and four-body \(\gamma^{(4)}\) multiple scattering (MS) terms [21]. Since Fe-C-N-Mn chains feature angles close to 180\({}^{\circ}\), the actual number of parameters used to define the three-body and the four-body peaks was reduced by symmetry. More details on the use of parameter correlation in the four-body term are out of the aim of the present work and can be found in references [22; 23].