Answering to the general question in the title of this post:
Assuming that by "stationarity" the OP means "second-order", "covariance-", "weak-" stationarity, i.e. the stationarity concept that relates to a constant mean and variance throughout a stochastic process, then the answer depends on whether the heteroskedasticity is conditional or unconditional.
If it is unconditional, it is incompatible with covariance-stationarity, because it negates it directly.
If it is conditional on certain co-variates, then, if these covariates are themselves stationary of the appropriate order (depending on how they enter the heteroskedastic function), the unconditional variance will be constant, and so conditional heteroskedasticity given such covariates is compatible with second-order stationarity.
But the above are relations at population-level.
Statistical tests are based on finite samples, and so the phenomenon where two statistical tests may provide conflicting indications is nothing new.