Two-sided-cell conjecture for integral quantizations

Let $\mathcal{D}$ be an integral quantization of a conical symplectic resolution, let $\mathcal{I}$ index the simple objects of its category $\mathcal{O}$, and define two indices to lie in the same two-sided cell using the left and right preorders. Let $\mathscr{S}^{\operatorname{sp}}$ be the set of special symplectic leaves, ordered by closure, and let $\mathfrak{M}_{\alpha,0}$ be the associated leaf closure. Two-sided-cell conjecture. The map

$$\{\text{two-sided cells}\}\longrightarrow\mathscr{S}^{\operatorname{sp}},\qquad [\alpha]\longmapsto\mathfrak{M}_{\alpha,0}$$

is an isomorphism of posets. The map is known to be surjective in general and bijective in hypertoric and certain integral flag-variety cases, but the full assertion remains open.