We report the results of continuous and time-resolved photoluminescence measurements on type-II GaSb quantum rings embedded within GaAs/Al_xGa_1-xAs quantum wells. A range of samples were grown with different well widths, compensation-doping concentrations within the wells, and number of quantum-ring layers. We find that each of these variants have no discernible effect on the radiative recombination, except for the very narrowest (5 nm) quantum well. In contrast, singleparticle numerical simulations of the sample predict changes in photoluminescence energy of up to 200 meV. This remarkable difference is explained by the strong Coulomb binding of electrons to rings that are multiply charged with holes. The resilience of the emission to compensation doping indicates that multiple hole occupancy of the quantum rings is required for efficient carrier recombination, regardless of whether these holes come from doping or excitation.