the SM case. To that end we construct maximally localized Wannier functions using the same four different, randomly chosen initial sets of Bloch phases as in Section \[sec:SM\]. Representative examples of CG optimization are shown in figure \[fig:cg\_results\_im\]. In contrast to the SM case, the localities of the resulting Wannier functions coincide for all four sets of phases. This is the case for all considered systems and polarizations, strongly indicating that the IM locality measure does not possess any local extrema.

![Locality ($\mathcal{I}_{n}$) of the IM-optimized Wannier functions as a function of GA generation for the Sq-D structure, TM polarization, 3rd band (top) and for the Sq-A structure, TE polarization, 5th band (bottom). The dashed green line shows the locality of the best-localized Wannier function in each generation. Every 100 generations these Wannier functions served as a starting point for the subsequent CG optimization step (red crosses). On the right-hand side, the modulus square of the SM-optimized Wannier functions is shown for an early (20000th) and a later (60000) generation, respectively.[]{data-label="fig:ga_im"}](all-g-tm-im "fig:"){width="0.95\columnwidth"}\
![Locality ($\mathcal{I}_{n}$) of the IM-optimized Wannier functions as a function of GA generation for the Sq-D structure, TM polarization, 3rd band (top) and for the Sq-A structure, TE 