Premet's orbit-fiber conjecture for finite-dimensional irreducible W-algebra modules

Let $\mathcal W$ be a finite W-algebra, let $\operatorname{Irr}_{fin}(\mathcal W)$ be the set of isomorphism classes of finite-dimensional irreducible $\mathcal W$-modules, let $\operatorname{Prim}_{\mathbb{O}}(\mathcal U)$ be the set of primitive ideals $\mathcal J\subset\mathcal U$ with $\operatorname{V}(\mathcal U/\mathcal J)=\overline{\mathbb{O}}$, and let $C(e)=Q/Q^\circ$ be the component group acting on $\operatorname{Irr}_{fin}(\mathcal W)$. For a module $N$, write $\operatorname{Ann}(N)^\dagger$ for the associated primitive ideal.

Premet's conjecture. The map

$$N\mapsto \operatorname{Ann}(N)^\dagger:\operatorname{Irr}_{fin}(\mathcal W)\rightarrow\operatorname{Prim}_{\mathbb{O}}(\mathcal U)$$

is surjective and each of its fibers is a single $C(e)$-orbit.

This conjecture describes finite-dimensional irreducible W-algebra modules in terms of primitive ideals with associated variety $\overline{\mathbb{O}}$, with the component group accounting precisely for the fibers. The supplied text gives no resolution status.