Feigin–Gainutdinov–Semikhatov–Tipunin equivalence conjecture for triplet $W$-algebras

Let $p\geq 2$, let $W(p)$ be the triplet vertex operator algebra, and let $\overline{U}_q(\mathfrak{sl}_2)$ be the restricted quantum enveloping algebra associated to $\mathfrak{sl}_2$ at a $2p$-th root of unity. Write $W(p)$-$\boldsymbol{\mathrm{mod}}$ for the category of $W(p)$-modules and $\overline{U}_q(\mathfrak{sl}_2)$-$\boldsymbol{\mathrm{mod}}$ for the category of finite-dimensional $\overline{U}_q(\mathfrak{sl}_2)$-modules. Feigin–Gainutdinov–Semikhatov–Tipunin conjecture. As a braided quasitensor category, $W(p)$-$\boldsymbol{\mathrm{mod}}$ is equivalent to $\overline{U}_q(\mathfrak{sl}_2)$-$\boldsymbol{\mathrm{mod}}$. This conjecture proposes a bridge between the representation theory of the logarithmic triplet conformal field theories and restricted quantum groups; the supplied text gives no evidence that the equivalence has been proved or disproved.