Lurie's conjecture on the twisted Whittaker category

Let $G$ be a reductive group, $G$ its Langlands dual group, and let $W^c(Gr_G):=D^cmod(Gr_G)^{N((t)),\chi}$ denote the twisted Whittaker category, where $c$ is the twisting parameter. Let $U_q(G)$ be the corresponding quantum group and set $q=\exp(i c)$. Lurie's conjecture. There exists an equivalence

$$W^c(Gr_G)\simeq Rep(U_q(G)),\qquad q=\exp(i c).$$

This conjecture proposes a quantum-group description of the twisted Whittaker category and is intended as a quantum deformation of the geometric Satake framework. The source presents it as Lurie's conjecture; no resolution status is supplied here.