  The $GW$ approximation has recently gained increasing attention as a viable
method for the computation of deep core-level binding energies as measured by
X-ray photoelectron spectroscopy (XPS). We present a comprehensive benchmark
study of different $GW$ methodologies (starting-point optimized, partial and
full eigenvalue-self-consistent, Hedin shift and renormalized singles) for
molecular inner-shell excitations. We demonstrate that all methods yield a
unique solution and apply them to the CORE65 benchmark set and ethyl
trifluoroacetate. Three $GW$ schemes clearly outperform the other methods for
absolute core-level energies with a mean absolute error of 0.3 eV with respect
to experiment. These are partial eigenvalue self-consistency, in which the
eigenvalues are only updated in the Green's function, single-shot $GW$
calculations based on an optimized hybrid functional starting point and a Hedin
shift in the Green's function. While all methods reproduce the experimental
relative binding energies well, the eigenvalue self-consistent schemes and the
Hedin shift yield with mean errors $<0.2$ eV the best results.
