Suppose $\kappa$ is strongly inaccessible and every stationary subset of $\kappa$ reflects. Must $\kappa$ be Mahlo?


Remarks:


1. It is possible for every stationary subset of $\kappa$ to reflect, but $\kappa$ is only weakly inaccessible (and not strongly inaccessible).
2. If $V=L$ then the answer is "yes", and in fact $\kappa$ must be weakly compact.



Does stationary reflection imply Mahloness?
Not necessarily. Here's a counterexample, but I'm sure it is a ridiculous overkill in consistency strength:


Suppose $\kappa$ is the least inaccessible limit of supercompact cardinals. Then $\kappa$ is not Mahlo. If $S \subseteq \kappa$ is stationary, then there is a stationary $T \subseteq S$ such that $T$ concentrates on some cofinality $\theta < \kappa$. Taking a supercompact $\delta \in (\theta,\kappa)$ gives the reflection of $T$ by the usual argument.


