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

Remarks:

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