Non-integral orbit-stabilizer conjecture for finite-dimensional W-algebra modules

Let $\mathbb{O}$ be a special nilpotent orbit, let $\lambda$ be a dominant representative of a regular central character, and let $W_\lambda$ be its integral Weyl group. Let $W_I$ be a standard parabolic subgroup of the affine Weyl group whose projection is conjugate to $W_\lambda$, let $\mathcal{D}_{\mathbb{O}}$ be the distinguished involutions in the affine two-sided cell corresponding to $\mathbb{O}$, and let $H_d\subset A(e)$ be the subgroup associated with $d\in\mathcal{D}_{\mathbb{O}}$. If $\mathcal J\in\operatorname{Pr}_{\mathbb{O}}(\mathcal U_\lambda)$ corresponds to $d\in W_I\cap\mathcal{D}_{\mathbb{O}}$, then the non-integral orbit-stabilizer conjecture asserts that the stabilizer of the $A(e)$-orbit in $\operatorname{Irr}_{fin}(\mathcal W)$ lying over $\mathcal J$ is

$$H_d\subset A(e).$$

The claim generalizes the integral orbit-stabilizer theorem. It is known in the extreme cases of regular and trivial nilpotent orbits, while compatibility with the integral theorem was unclear in the source.