Spherical Fundamental Local Equivalence

Let $G$ be a reductive group, let $\check{G}$ be its Langlands dual group, and let $\kappa$ and $\check{\kappa}$ be corresponding quantum parameters. Let $\operatorname{KL}_{\kappa}(G)$ denote the category defined above, let $\operatorname{Gr}_{\check{G}}$ be the affine Grassmannian, and let $N$ be the unipotent subgroup of a fixed Borel of $G$. For a non-degenerate character $\chi$, write $\operatorname{Whit}(\operatorname{DMod}_{\check{\kappa}}(\operatorname{Gr}_{\check{G}}))$ for the spherical Whittaker category, namely the strong $\mathcal{L}N$-invariants of $\operatorname{DMod}_{\check{\kappa}}(\operatorname{Gr}_{\check{G}})$. Spherical Fundamental Local Equivalence. For any level $\kappa$, there is an equivalence of unital factorization categories

$$\operatorname{KL}_{\kappa}(G) \simeq \operatorname{Whit}(\operatorname{DMod}_{\check{\kappa}}(\operatorname{Gr}_{\check{G}})).$$

This is an expected output of the 2-categorical formalism of the local geometric Langlands program. The supplied context does not indicate whether this expected equivalence has been proved or remains open.