Finkelberg–Mirković conjecture for general blocks

Let $J\subset S_{\mathrm{aff}}$ be finitary, let $C_J$ be the corresponding set of weights, and let $\lambda\in C_J$. Let $\mathsf{Perv}_{\mathrm{Whit},J}(\mathsf{Gr}^\circ,\Bbbk)$ be the indicated Whittaker-equivariant perverse-sheaf category, let $\mathsf{Rep}_{W_{\mathrm{aff}}\bullet\lambda}(\mathbf G)$ be the corresponding representation block, and let $\Delta_w^J$, $\nabla_w^J$, $\mathscr I\mathscr C_w^J$, $\mathsf M(w\bullet\lambda)$, $\mathsf N(w\bullet\lambda)$ and $\mathsf L(w\bullet\lambda)$ denote the standard, costandard, intersection-cohomology, Weyl, dual Weyl and simple objects. For finitary $J\subset K$, let the averaging functors be the geometric counterparts of the translation functors $T_\lambda^\mu$ and $T_\mu^\lambda$.

Finkelberg–Mirković conjecture. There exists an equivalence

$$\Psi_\lambda:\mathsf{Perv}_{\mathrm{Whit},J}(\mathsf{Gr}^\circ,\Bbbk)\xrightarrow{\sim}\mathsf{Rep}_{W_{\mathrm{aff}}\bullet\lambda}(\mathbf G)$$

that sends, for every $w\in W_{\mathrm{aff}}^J$, the three objects to the corresponding objects:

$$\Psi_\lambda(\Delta_w^J)\cong\mathsf M(w\bullet\lambda),\quad \Psi_\lambda(\nabla_w^J)\cong\mathsf N(w\bullet\lambda),\quad \Psi_\lambda(\mathscr I\mathscr C_w^J)\cong\mathsf L(w\bullet\lambda).$$

It intertwines the geometric-Satake action with the action of $\mathsf{Rep}(G/\mathrm Z(G))$. Moreover, the equivalences can be chosen so that, for finitary $J\subset K$, $\lambda\in C_J$ and $\mu\in C_K$, the translation functors correspond to the natural averaging functors.

This extends the conjecture of Finkelberg and Mirković from the principal block to general blocks. The supplied text does not state whether it has been resolved.