Donagi–Pantev's Dolbeault geometric Langlands conjecture

Let $G$ be a complex reductive group, let $X$ be a smooth complex projective curve, let $G^{\vee}$ be its Langlands dual group, and let $T^*Bun_G(X)$ and $T^*Bun_{G^{\vee}}(X)$ be the cotangent stacks of the two moduli stacks of bundles. Donagi–Pantev's Dolbeault geometric Langlands conjecture. There is an equivalence of dg categories

$$\mathcal QC(T^*Bun_G(X))\simeq \mathcal QC(T^*Bun_{G^{\vee}}(X)).$$

The conjecture is compatible with suitable Dolbeault Hecke functors and requires modification for singularities and non-compactness; the paper states that it was proved over a dense open locus.