Quantum geometric Langlands equivalence for twisted Whittaker and Kazhdan–Lusztig categories

Let $G$ be a reductive group with Langlands dual group $G$, and let $\kappa$ and $check\kappa$ be invariant forms on their Lie algebras such that, for the mutually dual Cartan subalgebras, the associated forms satisfy

$$B_{\check\fh}=B_{\fh}^{-1}.$$

Assume that the corresponding scalar $check c$ is not in $\mathbb Q^{\geq 0}$. Let $W^c(Gr_G)$ be the twisted Whittaker chiral category and $KL^{check\kappa}(G)$ the Kazhdan–Lusztig chiral category. Quantum geometric Langlands conjecture. There exists an equivalence of chiral categories

$$W^c(Gr_G)\simeq KL^{check\kappa}(G).$$

This is the Langlands-dual formulation obtained by combining the preceding conjectural equivalence with the factorizable-sheaves description. The paper explicitly notes that it is intended to hold even for rational, non-negative values of $c$; no proof or disproof is supplied.