Gaitsgory's quantum Langlands duality conjecture for affine Hecke categories

Let $\mathbb{G}$ and $check{\mathbb{G}}$ be Langlands-dual reductive groups with loop groups $LG$ and $L\check{G}$, Iwahori subgroups $I$ and $\check{I}$, and dual bilinear forms $\kappa$ and $\check{\kappa}$. The associated monodromic affine Hecke categories are

$$\operatorname{D-mod}_\kappa(I \backslash LG / I) \quad\text{and}\quad \operatorname{D-mod}_{-\check{\kappa}}(\check{I} \backslash L\check{G} / \check{I}).$$

Gaitsgory's conjecture. There is an equivalence of monoidal categories

$$\operatorname{D-mod}_\kappa(I \backslash LG / I) \simeq \operatorname{D-mod}_{-\check{\kappa}}(\check{I} \backslash L\check{G} / \check{I}).$$

This is the quantum Langlands duality prediction for affine Hecke categories. The source says that the statement was conjectured by Gaitsgory and presents it as the first goal of the section; its resolution is not established in the supplied material.