Lusztig's quotient conjecture for special pieces

Let $\mathfrak{g}$ be the Lie algebra of an exceptional simple algebraic group, let $\0$ be a special nilpotent orbit, and let ${\mathcal P}(\0)$ be its special piece. Let $H$ be the finite group arising in the quotient description of special pieces. Lusztig's quotient conjecture. Each special piece ${\mathcal P}(\0)$ is isomorphic to the quotient of a smooth variety $\widetilde{\mathcal{P}}(\0)$ by $H$:

$${\mathcal P}(\0)\cong\widetilde{\mathcal{P}}(\0)/H.$$

For classical types, Kraft and Procesi proved the analogous stronger quotient description by an elementary abelian $2$-group. The conjecture concerns exceptional Lie algebras and is described as matching the quotient construction obtained from the natural complement to $H$.