Freeness of divisors with equidimensional radical Jacobian ideal of depth two

Let $D \subseteq S$ be a divisor in a complex manifold $S$, defined at a point $p$ by $h \in \mathcal{O}_{S,p}$. Assume that its Jacobian ideal $J_h$ is radical, equidimensional, and has depth $2$ in $\mathcal{O}_{S,p}$. Depth-two freeness conjecture. The quotient $\mathcal{O}_{S,p}/J_h$ is Cohen--Macaulay; equivalently, such a divisor is free at $p$. This is posed as the expected behavior for radical Jacobian ideals of depth two; the surrounding discussion records special cases and examples but does not resolve the general assertion.