Sommers' non-normal orbit-closure conjecture for non-abelian Lusztig quotients

Let $\mathcal{O}$ be a nilpotent orbit, let $S_{\mathcal{O}}$ be its special piece, and let $\overline{A}(\mathcal{O})$ be its Lusztig quotient. For $\mathcal{O}'\in S_{\mathcal{O}}$, let $H(\mathcal{O}',\mathcal{O})$ be the Levi subgroup assigned by Lusztig. Sommers' non-normal orbit-closure conjecture. Suppose

$$\overline{A}(\mathcal{O})=S_3,\ S_4\text{ or }S_5,$$

and $H(\mathcal{O}',\mathcal{O})$ is neither $1$ nor $\overline{A}(\mathcal{O})$. Then there exist two distinct completely prime primitive ideals $J(\lambda_1)$ and $J(\lambda_2)$ such that

$$U(\mathfrak{g})/J(\lambda_1)|_{K_{\mathbb{C}}}\cong R(\mathcal{O}),\qquad U(\mathfrak{g})/J(\lambda_2)|_{K_{\mathbb{C}}}\cong R(\overline{\mathcal{O}}).$$

The surrounding discussion explains that this representation-theoretic prediction implies non-normality for the relevant orbit closures, and notes that the $G_2$ case of $\mathcal{O}_8$ has already been proved. The supplied text gives no general resolution of the conjecture.