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Split Conformal Prediction (SCP) provides a computationally efficient way to construct confidence intervals in prediction problems. Notably, most of the theory built around SCP is focused on the single test point setting. In real-life, inference sets consist of multiplepoints, which raises the question of coverage guarantees for many points simultaneously. While *on average*, the False Coverage Proportion (FCP) remains controlled, it can fluctuate strongly around its mean, the False Coverage Rate (FCR). We observe that when adataset is split multiple times, classical SCP may not control the FCP in a majority of the splits. We propose CoJER, a novel method that achieves sharp FCP control in probability for conformal prediction, based on a recent characterization of the distribution of conformal $p$-values in a transductive setting. This procedure incorporates an aggregation scheme which provides robustness with respect to modeling choices. We show through extensive real data experiments that CoJER provides FCP control while standard SCP does not. Furthermore, CoJER yields shorter intervals than the *state-of-the-art method* for FCP control and only slightly larger intervals than standard SCP.