1 0 1 X 3 2 1 0 1 2 3 Y 1 2 0 2 Y 2 Oracle 1 0 1 X 3 2 1 0 1 2 3 Y 1 2 0 2 Y 2 Bonferroni 1 0 1 X 3 2 1 0 1 2 3 Y 1 2 0 2 Y 2 M-CP 1 0 1 X 3 2 1 0 1 2 3 Y 1 2 0 2 Y 2 DR-CP 1 0 1 X 3 2 1 0 1 2 3 Y 1 2 0 2 Y 2 C-HDR 1 0 1 X 3 2 1 0 1 2 3 Y 1 2 0 2 Y 2 L-CP Figure 1: Prediction regions for a bivariate unimodal dataset, conditional on a unidimensional input. The black, green, and yellow contours represent regions with nominal coverage levels of 20%, 40%, and 80%, respectively. The figure is similar to Figure 2 in the main text, with Bonferroni added as a comparison. Both Bonferroni and M-CP are based on Conformal Quantile Regression (CQR) applied separately for each dimension. Table 1: Detailed metrics for the unimodal heteroscedastic process from Figure 1, with 1 − α fixed to 0.8. MC Median Size CEC- X CEC- Z WSC Test time Method ( × 100 ) ( × 100 ) Bonferroni 0.813 0.0036 9.07 0.15 0.0241 0.012 0.0249 0.0098 0.815 0.0063 0.00339 5.9e-05 M-CP 0.801 0.0037 8.62 0.074 0.0240 0.0031 0.0157 0.0031 0.796 0.012 0.0959 0.058 DR-CP 0.796 0.0019 6.83 0.042 0.432 0.019 0.403 0.015 0.697 0.0093 0.0557 0.00075 C-HDR 0.809 0.0025 6.97 0.039 0.0129 0.0059 0.0155 0.0037 0.815 0.0030 14.2 0.11 L-CP 0.798 0.0024 8.06 0.035 0.00586 0.00095 0.00549 0.0014 0.794 0.0039 0.0584 0.0012 1