The product of independent beta probabilities escalation (PIPE) design for dual-agent phase I dose-escalation trials is a Bayesian model-free approach for identifying multiple maximum tolerated dose combinations of novel combination therapies. Despite only being published in 2015, the PIPE design has been implemented in at least two oncology trials. However, these trials require patients to have completed follow-up before clinicians can make dose-escalation decisions. For trials of radiotherapy or advanced therapeutics, this may lead to impractically long trial durations due to late-onset treatment-related toxicities. In this paper, we extend the PIPE design to use censored time-to-event (TITE) toxicity outcomes for making dose escalation decisions. We show via comprehensive simulation studies and sensitivity analyses that trial duration can be reduced by up to 35%, particularly when recruitment is faster than expected, without compromising on other operating characteristics.