Equidimensionality conjecture for restricted nilpotent centralizer intersections

Let $G$ be a reductive group over a field $k$ of good characteristic, with Lie algebra ${\mathfrak g}$. Let ${\cal N}_1$ be the restricted nilpotent cone, let $e\in{\cal N}_1$, and let ${\mathfrak z}_{\mathfrak g}(e)$ be the centralizer of $e$. Equidimensionality conjecture. The intersection

$${\mathfrak z}_{\mathfrak g}(e)\cap{\cal N}_1$$

is equidimensional. The conjecture is motivated by examples where related structural results fail but this intersection nevertheless remains equidimensional; its general validity is left open.