Factorization Kazhdan–Lusztig conjecture for the Jacquet functor

Let $\hat{\mathfrak{g}}_{\kappa}\textup{-mod}^{G(\mathcal{O})}$ be the category of $G(\mathcal{O})$-equivariant modules for the affine Kac–Moody algebra at level $\kappa$, and let $\textup{Fact}(U^{\textup{Lus}}_q(\mathfrak{g})\textup{-mod})$ be the topological factorization category associated to the Lusztig quantum group. Under the Riemann–Hilbert correspondence, let the Jacquet functor be the factorizable functor from $(\hat{\mathfrak{g}}_{\kappa}\textup{-mod}^{G(\mathcal{O})})_{\textup{Ran}(X)}$ to the corresponding Riemann–Hilbert category. Factorization Kazhdan–Lusztig conjecture. The Jacquet functor is fully faithful, with its essential image identified with

$$\textup{Fact}(U^{\textup{Lus}}_q(\mathfrak{g})\textup{-mod})$$

under the Riemann–Hilbert correspondence. This is the proposed factorization form of the Kazhdan–Lusztig equivalence; the source describes it as conjectural and does not provide a resolution in the supplied text.