  We study the infrared part of the spectrum for UV-filtered staggered Dirac
operators and compare them to the overlap counterpart. With sufficient
filtering and at small enough lattice spacing the staggered spectra manage to
``mimic'' the overlap version. They show a 4-fold near-degeneracy, and a clear
separation between would-be zero modes and non-zero modes. This suggests an
approximate index theorem for filtered staggered fermions and a correct
sensitivity to the topology of QCD. Moreover, it supports square-rooting the
staggered determinant to obtain dynamical ensembles with $N_f=2$.
