Generalized semi-infinite cohomology formula at positive level

Fix a positive level $\kappa$, set $q=\operatorname{exp}\left(\frac{\pi\sqrt{-1}}{\kappa'-\kappa_{\textup{crit}}}\right)$, and let $M$ be a $U^{\textup{Lus}}_q(\mathfrak{g})$-module. For each $\mu\in\Lambda$, let $\mathfrak{C}^{\frac{\infty}{2}}_!(\mathfrak{n}(\mathcal{K}),M)^\mu$ denote the $\mu$-component of the !-generalized semi-infinite cohomology functor, and let $\textup{C}^{\bullet}(U^{\textup{KD}}_q(\mathfrak{n}),\textup{KL}^{\kappa}_G(M))^\mu$ denote the $\mu$-component of the derived $U^{\textup{KD}}_q(\mathfrak{n})$-invariants of the Kazhdan–Lusztig image of $M$. Generalized semi-infinite cohomology conjecture. For all $\mu\in\Lambda$,

$$\mathfrak{C}^{\frac{\infty}{2}}_!(\mathfrak{n}(\mathcal{K}),M)^\mu\cong\textup{C}^{\bullet}(U^{\textup{KD}}_q(\mathfrak{n}),\textup{KL}^{\kappa}_G(M))^\mu.$$

This is the conjectural !-analogue of the two formulas proved immediately beforehand for ordinary and !*-generalized semi-infinite cohomology. The supplied text gives no resolution of this formula.