The quasi-Hopf algebra correspondence for the triplet vertex operator algebra

Let $p\geq 2$, let $q=e^{\pi i/p}$, let $\overline{U}^{(\Phi)}_{q} s\ell(2)$ be the factorisable ribbon quasi-Hopf algebra described above, and let $\mathcal{W}(p)$ be the triplet vertex operator algebra. The triplet correspondence conjecture. There is an equivalence

$$\mathrm{\bf Rep}\,\overline{U}^{(\Phi)}_{q} s\ell(2)\cong \mathrm{\bf Rep}\,\mathcal{W}(p)$$

as ${\mathbb C}$-linear ribbon categories. This conjecture would identify the representation-theoretic and vertex-operator-algebraic realizations of the triplet theory; its resolution is not supplied here.