Koszul duality for the geometric embedding and Kazhdan–Laumon category O

Let $G$ be a semisimple group with Langlands dual group $G^\vee$, let $\mathfrak{u}_q$ be the small quantum group of $G^\vee$, and let $\overset{\bullet}{\mathfrak{u}}_q\mathrm{-mod}_{\hat{0}}$ denote the principal block of its category of modules with a compatible grading by the character lattice. Let $j_!:\mathcal{D}^{\mathrm{geom}}\to\widetilde{\mathcal{D}}_\psi$ be the functor obtained from the open embedding $j:G/B\hookrightarrow P_ -\backslash\mathcal{F}\ell$. Koszul-duality conjecture. The functor

$$j_!:\mathcal{D}^{\mathrm{geom}}\longrightarrow\widetilde{\mathcal{D}}_\psi$$

is Koszul dual to the functor

$$\mathrm{KL}_\mathcal{O}\longrightarrow\overset{\bullet}{\mathfrak{u}}_q\mathrm{-mod}_{\hat{0}}$$

constructed in Section 7 of the cited work. This conjecture is intended to relate the geometric embedding and the action of the Weyl group on the relevant categories to the Kazhdan–Laumon category $\mathrm{KL}_\mathcal{O}$ and the principal block of graded small-quantum-group modules; the source provides no resolution status.