Left-hand panels: Inferred PM values of M31 background QSOs in right ascension (${\mu }_{\alpha }^{\ast }$; left) and declination (μ_δ; right) as a function of G_BP − G_RP color. The upper sub-panels show the inferred PMs in each color bin for the full QSO sample (gray points), with the dashed gray line and shaded region indicating the overall zero-point offset and its 1σ uncertainty from the MCMC analysis. The lower sub-panels show results for QSOs with different astrometric solutions: blue and orange points correspond to the inferred PMs in each color bin for 5p and 6p sources, respectively. Dashed lines and shaded areas denote the overall PMs and their 1σ uncertainties for the two subsamples. Right-hand panels: Same as the left-hand panels, but for M33 background QSOs.