Kazhdan–Lusztig correspondence conjecture for $(1,p)$ logarithmic models

Let $p$ be the parameter of the $(1,p)$ logarithmic model, let $\mathcal{W}(p)$ denote its representation category, and let $\overline{\mathscr{U}}_{\mathfrak{q}}s\ell(2)$ be the restricted quantum group with

$$\mathfrak{q}=e^{\frac{i\pi}{p}}.$$

Kazhdan–Lusztig correspondence conjecture. The category of $\mathcal{W}(p)$-representations is equivalent to the category of finite-dimensional $\overline{\mathscr{U}}_{\mathfrak{q}}s\ell(2)$-representations. The conjecture concerns the categorical correspondence for the $(1,p)$ logarithmic models; unlike the semisimple case, it requires no semisimplification on the quantum-group side. The source states that this issue remains to be addressed.