This article develops hypothesis testing procedures for the stratified mark-specific proportional hazards model in the presence of missing marks. The motivating application is preventive HIV vaccine efficacy trials, where the mark is the genetic distance of an infecting HIV sequence to an HIV sequence represented inside the vaccine. The test statistics are constructed based on two-stage efficient estimators, which utilize auxiliary predictors of the missing marks. The asymptotic properties and finite-sample performances of the testing procedures are investigated, demonstrating double-robustness and effectiveness of the predictive auxiliaries to recover efficiency. The methods are applied to the RV144 vaccine trial.