SIMBAD references

Only a small fraction of gamma-ray bursts (GRBs) have independent redshift measurements, which are essential for understanding their intrinsic properties. For this reason, empirical correlations of GRBs have often been touted as useful distance indicators, for both individual GRBs as well as population studies. Building upon our previous work, we test the ability of the Yonetoku, 3D Dainotti, and the ${L}_{X}-{T}_{a}^{\ast }-{E}_{{\rm{iso}}}$ correlation to adequately constrain the GRB rate, Ψ_GRB. Our analysis demonstrates that, even under idealized conditions that neglect substantial uncertainties, the derived redshift solutions cannot accurately constrain Ψ_GRB, regardless of the intrinsic distribution's characteristic width. We thus demonstrate unequivocally that-notwithstanding the questionable assumption of no selection biases-empirical GRB correlations alone cannot serve as reliable distance indicators at either the individual or population level.