Even-rank associated-variety conjecture for type D affine vertex algebras

Let $r$ be an even integer, let $\mathfrak{g}$ be the type D Lie algebra used in the source, let $V_{2-r}(\mathfrak{g})$ be the simple affine vertex algebra at level $2-r$, and let $\mathbb{O}_{(2^{r-2},1^4)}$ denote the nilpotent orbit associated with the partition $(2^{r-2},1^4)$. For $f$ in this orbit, let $\mathcal{W}_{2-r}(\mathfrak{g},f)$ be the corresponding $W$-algebra. Even-rank associated-variety conjecture.

$$X_{V_{2-r}(\mathfrak{g})}=\overline{\mathbb{O}_{(2^{r-2},1^4)}}.$$

Consequently, $\mathcal{W}_{2-r}(\mathfrak{g},f)$ is lisse for $f\in\mathbb{O}_{(2^{r-2},1^4)}$. The source previously states this claim for the relevant even values of $r$, confirms it for $r=6$, and notes that the $r=4$ case follows from earlier work; it does not establish the full even-$r$ assertion in this paper.