Gaitsgory's dimension-torsor Whittaker equivalence conjecture

Let $\G$ be a loop group, let $\bN$ be its loop maximal unipotent subgroup, and let $\C$ be a category equipped with a $\bG$-action. Let $\chi$ be the character of $\bN$ defined from the simple roots, and let $\Theta_{\Lambda}:\C_{\bN,\chi}\to\C^{\bN,\chi}$ be the functor associated with a trivialization $\Lambda$ of the dimension torsor of $\bN$. Gaitsgory's Whittaker equivalence conjecture. For any trivialization of the dimension torsor of $\bN$, the functor

$$\Theta_{\Lambda}:\C_{\bN,\chi}\to\C^{\bN,\chi}$$

is an equivalence of categories.

This is the dimension-torsor-refined form of the assertion that Whittaker coinvariants and invariants agree. The source attributes the conjecture to Gaitsgory and gives no resolution in the stated generality.