The GW approximation has been recently gaining popularity among the methods for simulating molecular core-level X-ray photoemission spectra. Traditionally, Gaussian-type orbital GW core-level binding energies have been computed using either the cc-pVnZ or def2-nZVP basis set families, extrapolating the obtained results to the complete basis set limit, followed by an element-specific relativistic correction. Despite achieving rather good accuracy, it has been previously stated that these binding energies are chronically underestimated. In the present work, we show that those previous studies obtained results that were not well-converged with respect to the basis set size and that, once basis set convergence is achieved, there seems to be no such underestimation. Standard techniques known to offer a good cost-accuracy ratio in other theories demonstrate that the cc-pVnZ and def2-nZVP families exhibit contraction errors and might lead to unreliable complete basis set extrapolations for absolute binding energies, often deviating about 200-500 meV from the putative basis set limit found in this work. On the other hand, uncontracted versions of these basis sets offer vastly improved convergence. Even faster convergence can be obtained using core-rich property-optimized basis set families like pcSseg-n, pcJ-n, and ccX-nZ. Finally, we also show that the improvement observed for core properties using these specialized basis sets does not degrade their description of valence excitations: vertical ionization potentials and electron affinities computed with these basis sets converge as fast as the ones obtained with the aug-cc-pVnZ family, thus offering a balanced description of both core and valence regions.