Gaitsgory–Lurie spherical Whittaker–affine Lie algebra equivalence conjecture

Let $G$ be a reductive group, let $\check{G}$ be its Langlands dual group, and let $\kappa$ be a nondegenerate level for $G$ with dual level $\check{\kappa}$ for $\check{G}$. Write $\operatorname{Gr}_G=\mathfrak{L}G/\mathfrak{L}^+G$ for the affine Grassmannian, and let $\mathsf{Whit}_{\kappa}^{\operatorname{sph}}$ be the DG category of $\kappa$-twisted Whittaker $D$-modules on $\operatorname{Gr}_G$. Let $\widehat{\check{\mathfrak{g}}}_{\check{\kappa}}$ be the affine Lie algebra associated to $\check{G}$ and $\check{\kappa}$, and let $\widehat{\check{\mathfrak{g}}}_{\check{\kappa}}\text{\textendash}\mathsf{mod}^{\mathfrak{L}^+\check{G}}$ denote the DG category of Harish–Chandra modules for $(\widehat{\check{\mathfrak{g}}}_{\check{\kappa}},\mathfrak{L}^+\check{G})$. Gaitsgory–Lurie conjecture. There is an equivalence of DG categories

$$\mathsf{Whit}_{\kappa}^{\operatorname{sph}}\simeq \widehat{\check{\mathfrak{g}}}_{\check{\kappa}}\text{\textendash}\mathsf{mod}^{\mathfrak{L}^+\check{G}}.$$

This is a spherical form of the local quantum geometric Langlands conjecture, relating twisted Whittaker sheaves on the affine Grassmannian to Harish–Chandra modules for the dual affine Lie algebra. The supplied text does not state whether the conjecture is known or remains open.