Quantum geometric Langlands conjecture

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 $\mathfrak{\det}$ denote the determinant line bundle, let $\mathcal D_k(Bun_G(X))$ and $\mathcal D_{k^{\vee}}(Bun_{G^{\vee}}(X))$ be the categories of twisted $\mathcal D$-modules at levels $k$ and $k^{\vee}$, and impose $k^{\vee}=-1/k$. Quantum geometric Langlands conjecture. There is an equivalence

$$\mathcal D_k(Bun_G(X))\simeq \mathcal D_{k^{\vee}}(Bun_{G^{\vee}}(X)),\qquad k^{\vee}=-1/k.$$

The conjecture is a quantized form of de Rham geometric Langlands; the source explains its expected Hecke/localization compatibility and its degeneration to the usual de Rham conjecture as $k\to0$ and $k^{\vee}\to\infty$.