  The dependence of ab initio many-body perturbation theory within the $GW$
approximation on the eigensystem used in calculating quasiparticle corrections
limits this method's predictive power. Here, we investigate the accuracy of the
recently developed Wannier-localized optimally tuned screened range-separated
hybrid (WOT-SRSH) functional as a generalized Kohn-Sham starting point for
single-shot $GW$ ($G_0W_0$) calculations for a range of semiconductors and
insulators. Comparison to calculations based on well-established functionals,
namely PBE, PBE0, and HSE, as well as to self-consistent $GW$ schemes and to
experiment, shows that band gaps computed via $G_0W_0$@WOT-SRSH have a level of
precision and accuracy that is comparable to that of more advanced methods such
as quasiparticle self-consistent $GW$ (QS$GW$) and eigenvalue self-consistent
$GW$ (ev$GW$). We also find that $G_0W_0$@WOT-SRSH improves the description of
states deeper in the valence band manifold. Finally, we show that
$G_0W_0$@WOT-SRSH significantly reduces the sensitivity of computed band gaps
to ambiguities in the underlying WOT-SRSH tuning procedure.
