Gaitsgory–Lysenko–Drinfeld–Stoyanovsky quantum geometric Langlands conjecture

Let $G$ be a reductive group, let $\check{G}$ be its Langlands dual group, and let $\kappa$ be a Kac–Moody level for $G$. Denote by $\check{\kappa}$ the dual level for $\check{G}$, by $\operatorname{Bun}_G$ and $\operatorname{Bun}_{\check{G}}$ the corresponding moduli stacks of bundles, and by $\operatorname{DMod}_{\kappa}$ the category of twisted $D$-modules at level $\kappa$. Gaitsgory–Lysenko–Drinfeld–Stoyanovsky conjecture. Quantum geometric Langlands predicts an equivalence

$$\operatorname{DMod}_{\kappa}(\operatorname{Bun}_G) \xrightarrow[\mathbb L_{\kappa}]{\sim} \operatorname{DMod}_{-\check{\kappa}}(\operatorname{Bun}_{\check{G}}).$$

This is the expected quantum geometric Langlands equivalence. The source gives the prediction without asserting a proof or describing a known range of cases.