Finite-dimensionality criterion for highest weight modules of finite W-algebras

Let $ab ab$ be a $W_0$-orbit in $ab^*$, and choose $ab 't\inab$ such that

$$(\lambda|\alpha^\vee)\notin\mathbb Z_{>0}$$

for every $ab\in\Phi_0^+$. Let $L(\Lambda,e)$ be the irreducible highest weight $U(\mathfrak g,e)$-module associated with $ab$, and let $\mathcal{V\!A}$ denote the associated variety of an annihilator ideal. Finite-dimensionality criterion. The module $L(\Lambda,e)$ is finite dimensional if and only if

$$\mathcal{V\!A}(\operatorname{Ann}_{U(\mathfrak g)}(L(\lambda)))=\overline{G\cdot e}.$$

The paper states that this conjecture is verified in type $A$ for the standard choice of positive roots; its status in general is therefore not resolved by the supplied text.