these elements. The disappearance of these elements in other molecules (e.g. CaOH) could also explain the observed discrepancy. We note that the potassium doublet at around 760/790nm is well reproduced by the models, except for the latest spectral types ($\geq$M7) and the coolest sources (Figs. \[fig\_MplusTeff\_sdM:full\_XSH\_model\_lines\_sdM\], \[fig\_MplusTeff\_esdM:full\_XSH\_model\_lines\_esdM\], \[fig\_MplusTeff\_usdM:full\_XSH\_model\_lines\_usdM\]).

As a consequence, we opted for four fitting procedures, described below, to gauge the uncertainties on the physical parameters derived from the synthetic spectra. We describe them below:

1.  “FF” corresponds to the fit of the full SED of each subtype and metal class from 450 to 2500 nm (Figs. \[fig\_MplusTeff\_sdM:full\_XSH\_model\_FF\_sdM\]–\[fig\_MplusTeff\_sdM:full\_XSH\_model\_FF\_usdM\]).

2.  “LL” corresponds the fitting procedure of a few lines in the optical spectra (K and Na). The model spectra shown correspond to the physical parameters derived from the line fits (Figs. \[fig\_MplusTeff\_sdM:full\_XSH\_model\_LL\_sdM\]–\[fig\_MplusTeff\_sdM:full\_XSH\_model\_LL\_usdM\]).

3.  “FL”: in this case we fix the temperature derived from the full fit and adjust gravity and metallicity to converge towards the best fit of the aforementioned lines (Figs. \[fig\_MplusTeff\_sdM:full\_XSH\_model\_FL\_sdM\]–\[fig\_MplusTeff\_sdM:full\_XSH\_model\_FL\_usdM\]).

4.  “LF”: we fix the gravity and metallicity from the fits of the line and adjust the effective temperature fitting the full SED of each subtype and subclass. (Figs. \[fig\_MplusTeff\_sdM:full\_XSH\_model\_LF\_sdM\]–\[fig\_MplusTeff\_sdM:full\_XSH\_model\_LF\_usdM\]).

In general, we find that the “FF” 