M_BH vs. D_d with 1σ uncertainties based on the ideal kinematic data. This provides a clearer visualization of the upper-left panel in Fig. 5. The plot also illustrates how the BH mass affects the inference of D_d, given that β_ani is within the prior range of −0.3 to 0.3. The gray shaded region represents models with the BH mass satisfying $f_{\mathrm{BIC}}^{\ast }\ge 0.2$, indicating a significant contribution to D_d when combining all models together.