Betti geometric Langlands for nilpotent sheaves

Let $G$ be a complex reductive group, let $X$ be a smooth complex projective curve, and let $G^{\vee}$ be its Langlands dual group. Let $\mathcal Shv_{\mathcal N}(Bun_G(X))$ be the dg category of sheaves with singular support in the global nilpotent cone, and let $\mathcal Loc_{G^{\vee}}(X)$ be the Betti moduli stack of local systems. Betti geometric Langlands for nilpotent sheaves. There is an equivalence

$$\mathcal Shv_{\mathcal N}(Bun_G(X))\simeq \mathcal QC^!_{\mathcal N}(\mathcal Loc_{G^{\vee}}(X))$$

compatible with Hecke functors. This is explicitly described as an initial rough form of the Betti geometric Langlands conjecture; the later formulation adds the expected compatibility with Wilson-line actions.