Kazhdan–Lusztig correspondence conjecture for logarithmic conformal field theories

Let $\mathcal{W}^{\log}$ be the proposed logarithmic conformal field theory, and let the smaller long-screening kernel be the corresponding subalgebra. Kazhdan–Lusztig correspondence conjecture. The representation category of $\mathcal{W}^{\log}$ should be a non-semisimple modular tensor category, equivalent as an abelian category to the representation category of the corresponding quantum group or Drinfeld double of the Nichols algebra, and equivalent as a modular tensor category to the representation category of a quasi-Hopf algebra generalizing the quantum group. The mapping-class-group action of $\mathrm{SL}_2(\mathbb{Z})$ on VOA representations should coincide with the action on the quantum-group center. Moreover, the representation category of the smaller long-screening kernel should be equivalent to that of Lusztig's infinite divided-power quantum group, or more generally the post-Nichols algebra. These categorical and modular equivalences are presented as conjectural; no resolution is given.