Inference for the parameters indexing Cox regression models is routinely based on the assumption that the model is correct and a priori specified. This is unsatisfactory because the chosen model is usually the result of a data-adaptive model selection process, which induces bias and excess uncertainty that is not usually acknowledged; moreover, the assumptions encoded in the resulting model rarely represent some a priori known, ground truth. Standard inferences may therefore lead to bias in effect estimates, and may moreover fail to give a pure reflection of the information that is contained in the data. Inspired by developments on assumption-free inference for so-called projection parameters, we here propose nonparametric definitions of main effect estimands which reduce to standard main effect parameters in Cox regression models when these models are correctly specified, but continue to capture the primary (conditional) association between a variable and an event time, even when these models are misspecified. We achieve an assumption-lean inference for these estimands by deriving their influence curve under the nonparametric model and invoking flexible data-adaptive algorithms.