Premet's uniqueness conjecture for primitive ideals of W-algebras

Let $\mathfrak g$ be a reductive Lie algebra, let $e\in\mathfrak g$ be nilpotent with orbit $\mathbb O$, and let $\mathcal U$ and $\mathcal W$ be the corresponding enveloping algebra and W-algebra. Choose an $\mathfrak{sl}_2$-triple $(e,h,f)$ and set $Q=Z_G(e,h,f)$, with component group $C(e)=Q/Q^\circ$. Let $\mathfrak{Id}(\mathcal U)$ and $\mathfrak{Id}_{\mathrm{fin}}(\mathcal W)$ denote the sets of two-sided ideals of $\mathcal U$ and finite-codimensional two-sided ideals of $\mathcal W$, respectively, and let $\mathcal I\mapsto\mathcal I^\dagger$ be the map from ideals of $\mathcal W$ to ideals of $\mathcal U$. Define

$$\mathfrak{Id}_{\mathbb O}(\mathcal U)=\{\mathcal J\in\mathfrak{Id}(\mathcal U)\mid \operatorname{V}(\mathcal U/\mathcal J)=\overline{\mathbb O}\}.$$

Premet's uniqueness conjecture. For any primitive $\mathcal J\in\mathfrak{Id}_{\mathbb O}(\mathcal U)$, the set of primitive ideals $\mathcal I\in\mathfrak{Id}_{\mathrm{fin}}(\mathcal W)$ satisfying $\mathcal I^\dagger=\mathcal J$ is a single $C(e)$-orbit.

This strengthens the existence conjecture by asserting uniqueness up to the component-group action. The supplied text does not state whether this conjecture has been proved or disproved.