Frenkel–Gaitsgory's affine Beilinson–Bernstein localization conjecture at the critical level

Let $G$ be the group under consideration, with Langlands dual group $\check{G}$, and let $K=k((t))$. Write $D_{crit}(\operatorname{Gr}_G)$ for the critical-level DG category of $D$-modules on the affine Grassmannian, $\operatorname{Op}_{\check{G}}^{reg}$ for the space of regular $\check{G}$-opers, and $\widehat{\mathfrak g}_{crit}\text{\textendash}\operatorname{mod}_{reg}$ for the DG category of regular critical-level $\widehat{\mathfrak g}$-modules. The geometric Satake action gives a $\mathsf{Rep}(\check{G})$-action on $D_{crit}(\operatorname{Gr}_G)$, and the canonical map $\operatorname{Op}_{\check{G}}^{reg}\to\mathbb B\check{G}$ gives a symmetric monoidal functor $\mathsf{Rep}(\check{G})\to\mathsf{QCoh}(\operatorname{Op}_{\check{G}}^{reg})$.

Frenkel–Gaitsgory's main conjecture. The induced functor

$$\Gamma^{\operatorname{Hecke}}:D_{crit}(\operatorname{Gr}_G)\underset{\mathsf{Rep}(\check{G})}{\otimes}\mathsf{QCoh}(\operatorname{Op}_{\check{G}}^{reg})\to\widehat{\mathfrak g}_{crit}\text{\textendash}\operatorname{mod}_{reg}$$

is a $t$-exact equivalence of DG categories.

This conjecture asserts that the obstructions to critical-level localization are precisely accounted for by regular central characters and the compatibility between the geometric Satake action and the oper action. Its resolution is not supplied in the source.