Smoothness of the normalization of a free divisor with radical Jacobian ideal

Let $D \subseteq S$ be a divisor in a complex manifold $S$ that is locally at a point $p$ given by $h=0$, and let $\pi:\widetilde D \rightarrow D$ be its normalization. Suppose that $D$ is free at $p$ and that $J_h=\sqrt{J_h}$. Normalization smoothness conjecture. The normalization $\widetilde D$ is smooth at $\pi^{-1}(p)$. The authors do not know whether the normalization hypothesis in the preceding theorem is necessary; this conjecture would provide an affirmative answer under the stated freeness and radical-Jacobian assumptions.