Suppose that $S$ is a compact smooth 2-dimensional manifold embedded in $\mathbb{R}^n$, where $n>2$. The medial axis of $S$ is the set of all points in $\mathbb{R}^n$ that have more than one closest point to $S$. Let $M$ denote the medial axis of $S$. I believe that $M$ has Hausdorff dimension at most $n-1$, as shown by Erdős in this paper: On the Hausdorff dimension of some sets in Euclidean space.
My question is: is it true that if $p \in M$ is a generic point on the medial axis, then in the neighborhood of $p$, $M$ must itself be a smooth manifold? I.e., can the medial axis be decomposed into a finite (or countable) union of smooth $d-1$ dimensional manifolds? I'm willing to make assumptions on the embedding of the manifold $S$, such as it being "generic" in a suitable sense, if necessary.
It seems that the statement holds if the embedded manifold is subanalytic, as shown in this reference. In this case, the medial axis is also subanalytic and the medial axis can be stratified. I'm wondering if the same can be said if the manifold is smooth.