Character-sheaf central functor equivalence conjecture

Let ${\mathcal C}_\zeta^{{\underline c}_f}$ be the monoidal category associated with a finite two-sided cell ${\underline c}_f\subset W_f(\zeta)$, and let ${\mathcal C}^\zeta_G(G)^{{\underline c}_f}$ be the corresponding summand of the category of character sheaves with central character $\zeta$. The commutator functor ${\mathfrak G}:{\mathcal C}_\zeta^{{\underline c}_f}\to{\mathcal C}^\zeta_G(G)^{{\underline c}_f}$ has a right adjoint that factors through a central functor

$${\mathfrak F}:{\mathcal C}^\zeta_G(G)^{{\underline c}_f}\to Z({\mathcal C}_\zeta^{{\underline c}_f}),$$

where $Z({\mathcal C}_\zeta^{{\underline c}_f})$ is the Drinfeld center. Character-sheaf central functor equivalence conjecture. The functor ${\mathfrak F}$ is an equivalence of categories.

This predicts that the relevant character-sheaf summand identifies with the Drinfeld center of the cell category. The supplied passage gives the construction of the functor and its central structure, but does not state whether the equivalence is known or open.