Several new and improved methods have been developed for the reconstruction of local potentials from electron densities (i.e., for the inversion of the Kohn-Sham equations), possibly also using orbitals, density matrices or wavefunctions . For an overview and discussions of successes and challenges, we refer to the reviews by Wasserman and coworkers . The application of potential reconstruction algorithms in sDFT was reviewed by Banafsheh and Wesolowski A comparative analysis of several reconstruction techniques for use in sDFT/FDE revealed a number of interesting conclusions for practical applications. In particular, it turned out that the residual density error of typical standard reconstruction schemes can be reduced by combining them. For instance, combining a first reconstruction step with an efficient basis-set representation of the reconstructed potential with a numerically more flexible grid-based scheme considerably reduced the resulting error. Moreover, a comparison of top-down reconstruction, in which a supersystem calculation is needed as a reference, and bottom-up reconstruction, in which the supersystem density is obtained as a sum of subsystem densities, was carried out in that work. It could be concluded that bottom-up embedding typically leads to more accurate results for densities, but is less robust for more strongly interacting systems, where top-down embedding is preferable.