In regression, conformal prediction often suffers from inefficiency under heteroscedasticity and skewness due to fixed, non-adaptive interval centering. We propose CoCP, a new method that parameterizes prediction intervals by a center $m(x)$ and a radius $h(x)$, and alternates between learning $h(x)$ via smooth quantile regression on folded residuals and refining $m(x)$ using a smooth interval loss. This corrects mis-centering and drives the interval toward high-density regions. Beyond finite-sample marginal validity via split-conformal calibration, we prove that CoCP also attains asymptotic conditional coverage and optimal interval length if the base estimators are consistent. Experiments demonstrate that CoCP yields tighter intervals and achieves improved empirical conditional-coverage reliability. Code is available at https://github.com/jjoeysu/cocp.

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