Logarithmic Kazhdan–Lusztig correspondence for Feigin–Tipunin vertex algebras

Let $\mathfrak{g}$ be a semisimple finite-dimensional complex Lie algebra with simple roots $\alpha_1,\ldots,\alpha_n$ and Killing form $(-,-)$. For an integer $p\geq 2$ divisible by the lacity of $\mathfrak{g}$, let $\Lambda=\sqrt{p}\Lambda^\vee$, let $\mathcal{W}_p(\mathfrak{g})$ be the kernel of the screening operators, and set $q=e^{\pi\mathrm{i}/p}$. Let $\mathfrak{B}(q)$ be the Nichols algebra generated by the screening operators, and let $\mathcal{Z}_\mathcal{C}(\operatorname{Mod}(\mathfrak{B}(q)))$ denote the corresponding Drinfeld center in the braided category $\mathcal{C}$. Logarithmic Kazhdan–Lusztig correspondence. There is an equivalence of braided tensor categories

$$\operatorname{Rep}(\mathcal{W}_p(\mathfrak{g}))\cong \mathcal{Z}_\mathcal{C}(\operatorname{Mod}(\mathfrak{B}(q))).$$

This correspondence identifies representations of the logarithmic Feigin–Tipunin vertex algebra with representations of the associated full quasi-quantum group. The case $\mathfrak{sl}_2$ is proven, while the general statement is presented conditionally and remains open.