The rough form of the Betti Geometric Langlands correspondence

Let $E$ be a field of characteristic zero. Let $G$ be a reductive group with Langlands dual group $G^{\vee}$, let $X$ be the underlying curve, and let $\textup{Loc}_{G^{\vee}}(X)_{E}$ be the derived stack of $G^{\vee}$-local systems on $X$ over $E$. Let $\textup{Ind}\operatorname{Coh}_{\mathcal N}(\textup{Loc}_{G^{\vee}}(X)_{E})$ denote ind-coherent sheaves with nilpotent singular support, and let $\mathit{Sh}_{\mathcal N_G(X)}(\textup{Bun}_G(X),E)$ denote the sheaf category on the moduli stack of $G$-bundles with nilpotent singular support. Rough form of the Betti Geometric Langlands correspondence. There is an equivalence

$$\textup{Ind}\operatorname{Coh}_{\mathcal N}(\textup{Loc}_{G^{\vee}}(X)_{E}) \simeq \mathit{Sh}_{\mathcal N_G(X)}(\textup{Bun}_G(X),E)$$

compatible with Hecke modifications and parabolic induction. This is a foundational expected form of the Betti Geometric Langlands correspondence; the source presents it as a rough statement of the correspondence rather than indicating that it has been proved.