Choose Between CQR and Plain Conformal for Heteroscedastic Demand

A marketplace forecasts order value. Noise grows with the predicted value: small baskets have a standard deviation around 0.5, large ones around 2.3. A data scientist wraps the mean model in split conformal prediction at 90 percent and gets one width, 5.6, for every order. A reviewer asks for conformalized quantile regression instead.

1. Explain what plain conformal gets right and what it gets wrong in this setting, using the numbers: coverage by order-size bucket was 1.000, 0.987, 0.927, 0.825, 0.723 from smallest to largest, 0.892 overall.
2. Describe CQR step by step, including the nonconformity score. Work this tiny example: \alpha=0.2, nine calibration scores E=[-1.2,-0.8,-0.5,-0.3,-0.1,0.0,0.2,0.4,1.5], and at a new x the quantile models give a lower bound 12 and an upper bound 20. What interval does CQR return? What would it mean if the 8th smallest score had been -0.2?
3. In the lesson simulation, CQR had mean width 5.32 against 5.61 and bucket coverage 0.937, 0.922, 0.893, 0.902, 0.899. Does CQR therefore guarantee coverage within each bucket? What does it guarantee?
4. Which would you ship, and what would you audit?