If $\kappa>\omega_1$ is a regular cardinal, $FP_\kappa$ is the assertion that every stationary subset of $\kappa$ consisting of ordinals of countable cofinality has a closed subset of order type $\omega_1$.
Question: If $\kappa$ is supercompact or has a supercompact cardinal below it, is it consistent that $FP_\kappa$ fails?
For context: I'm trying to show that a particular strengthening of PFA does not imply MM. Friedman's property (which under MM holds at almost all regular cardinals) seems like a natural candidate for separating them. My understanding is that the standard way of forcing a failure of FP is to add a nonreflecting stationary set; unfortunately, this preserves neither my axiom nor supercompactness. It is not too hard to show that if $FP_\kappa$ fails at a supercompact cardinal $\kappa$ (or at some larger cardinal), it still holds after performing the iteration to force the axiom, but I'm having trouble showing that that premise is consistent.