The quantum Betti global geometric Langlands conjecture

Let $G$ be a reductive group, let $\check{G}$ be its Langlands dual group, and let $\operatorname{Shv}_{c,\operatorname{Nilp}}(\operatorname{Bun}_G)$ denote the category of $c$-twisted sheaves on $\operatorname{Bun}_G$ with nilpotent singular support. Let $\int_X\operatorname{Rep}_q(\check{G})$ denote the corresponding factorization or topological field-theoretic category over $X$. The quantum Betti global geometric Langlands conjecture. There exists a canonical equivalence of derived categories

$$\operatorname{Shv}_{c,\operatorname{Nilp}}(\operatorname{Bun}_G)\xrightarrow[\cong]{\mathbb L_c^{\operatorname{Betti}}}\int_X\operatorname{Rep}_q(\check{G}).$$

This is the Betti analogue of the quantum global geometric Langlands correspondence and is motivated by the Betti Langlands correspondence and topological field theory; its general validity remains open.