Non-Cohen–Macaulay invariant rings and normality of the dual nilpotent cone

Let $G$ be a reductive group over a field of positive characteristic $p$, let $\mathfrak{h}$ be a Cartan subalgebra with Weyl group $W$, and let $\mathcal{N}^*\subseteq \mathfrak{g}^*$ be the dual nilpotent cone. Suppose that the invariant ring $S(\mathfrak{h})^W$ is not Cohen–Macaulay.

Non-normality question. Is the dual nilpotent cone $\mathcal{N}^*$ not a normal variety?

The question asks whether failure of the Cohen–Macaulay property for the invariant ring obstructs normality of the dual nilpotent cone. The paper notes that non-Cohen–Macaulay invariant rings occur in cases such as $(E_7,3)$, $(E_8,3)$, and $(E_8,5)$, but does not resolve the question in general.