Existing conformal prediction methods for time-to-event outcomes leverage only baseline covariates, producing prediction intervals that are insufficiently informative to facilitate decision making. We propose History-Aware Prediction Sets (HAPS), a conformal framework that constructs prediction sets for individual event times using covariate histories observed up to a decision time, targeting coverage among individuals who have survived to this time. HAPS handles right censoring adjusted for time-varying confounders via inverse probability of censoring weighting. When the censoring weights are consistently estimated, it achieves PAAC (probably asymptotically approximately correct) coverage among survivors. We further propose two doubly robust extensions of HAPS to weaken reliance on consistent estimation of the censoring distribution. In simulations, HAPS and its extensions reduce median prediction interval length by up to 75\% relative to baseline comparators while maintaining close to nominal coverage. On two public benchmark data sets, HAPS reduces the median interval length by up to 60\% for predictions at year 5, compared to the baseline comparators.

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