  The multi-tier $GW$+EDMFT scheme is an ab-initio method for calculating the
electronic structure of correlated materials. While the approach is free from
ad-hoc parameters, it requires a selection of appropriate energy windows for
describing low-energy and strongly correlated physics. In this study, we test
the consistency of the multi-tier description by considering different
low-energy windows for a series of cubic SrXO$_3$ (X=V,Cr,Mn) perovskites.
Specifically, we compare the 3-orbital $t_{2g}$ model, the 5-orbital
$t_{2g}+e_g$ model, the 12-orbital $t_{2g}+O_p$ model, and (in the case of
SrVO$_3$) the 14-orbital $t_{2g}+e_g+O_p$ model and compare the results to
available photoemission and X-ray absorption measurements. The multi-tier
method yields consistent results for the $t_{2g}$ and $t_{2g}+e_g$ low-energy
windows, while the models with $O_p$ states produce stronger correlation
effects and mostly agree well with experiment, especially in the unoccupied
part of the spectrum.
  We also discuss the consistency between the fermionic and bosonic spectral
functions and the physical origin of satellite features, and present
momentum-resolved charge susceptibilities.
