We develop a framework to quantify predictive uncertainty in survival analysis, providing a reliable lower predictive bound (LPB) for the true, unknown patient survival time. Recently, conformal prediction has been used to construct such valid LPBs for type-I right-censored data, with the guarantee that the bound holds with high probability. Crucially, under the type-I setting, the censoring time is observed for all data points. As such, informative LPBs can be constructed by framing the calibration as an estimation task with covariate shift, relying on the conditionally independent censoring assumption. This paper expands the conformal toolbox for survival analysis, with the goal of handling the ubiquitous general right-censored setting, in which either the censoring or survival time is observed, but not both. The key challenge here is that the calibration cannot be directly formulated as a covariate shift problem anymore. Yet, we show how to construct LPBs with distribution-free finite-sample guarantees, under the same assumptions as conformal approaches for type-I censored data. Experiments demonstrate the informativeness and validity of our methods in simulated settings and showcase their practical utility using several real-world datasets.