The problem of how to best select variables for confounding adjustment forms one of the key challenges in the evaluation of treatment effects in observational studies, and has been the subject of vigorous recent activity in causal inference. A major drawback of routine procedures is that there is no finite sample size at which they are guaranteed to deliver treatment effect estimators and associated confidence intervals with adequate performance. In this talk, I will consider this problem in the context of estimation of the conditional causal hazard ratio from observational studies under the assumption of no unmeasured confounding. I will propose tests of the null hypothesis that the exposure has no effect on the considered survival endpoint, which are uniformly valid under specific sparsity conditions, along with corresponding uniformly valid confidence intervals. Such uniform validity will be achieved by relying on penalized bias-reduced estimation procedures. I will discuss the implications of the proposal for variable selection in randomised experiments with survival endpoints, and show empirical evidence from simulation studies and real data analyses.