Suppose $U$ is a normal ultrafilter on $\kappa$ of Mitchell order zero, and let $j_U : V \to M$ be the associated embedding. Does there exist a nonstationary $X \subseteq \kappa^+$ such that $X \in M$ and $M \models X$ is stationary?

Note that if $W$ is a normal measure derived from an embedding $i : V \to N$ where $\mathcal P(\kappa^+) \subseteq N$, then $W$ gives a stationary-correct ultrapower, hence the restriction to low Mitchell order.