The quantum Langlands equivalence for the metaplectic symplectic group

Let $X$ be a smooth projective curve, let $n\geq 1$, let $G=\operatorname{Sp}(M_n)$ with $M_n={\cal O}_X^n\oplus\Omega^n$, and let $H=\operatorname{SO}_{2n+1}$. Write $\operatorname{Bun}_H$ for the moduli stack of $H$-bundles and $\operatorname{\widetilde{Bun}}_G$ for the metaplectic gerbe over $\operatorname{Bun}_G$. Let $\operatorname{D}_-(\operatorname{\widetilde{Bun}}_G)$ be the derived category on which the metaplectic involution acts by $-1$, and let $\bar\epsilon$ denote the automorphism induced by $-1$ on the symplectic bundle.

Quantum Langlands conjecture. There is an equivalence

$$QL: \operatorname{D}(\operatorname{Bun}_H)\\,\widetilde\to\\,\operatorname{D}_-(\operatorname{\widetilde{Bun}}_G)$$

\ncommuting with the actions of $\operatorname{Rep}(\operatorname{Sp}_{2n})$ by Hecke functors. Under this equivalence, the grading of $\operatorname{D}(\operatorname{Bun}_H)$ by the connected components of $\operatorname{Bun}_H$ corresponds to the grading of $\operatorname{D}_-(\operatorname{\widetilde{Bun}}_G)$ by the action of $\bar\epsilon$.

This is the specialization of the quantum Langlands conjecture to the dual pair $(H,G)$; the statement is presented as conjectural in the source, and no resolution is supplied.