The binary polyhedral groups conjecture for the orbifold VOA category

Let $L_2$ be the $A_1$ root lattice and let $V:=V_{L_2}$ be the corresponding lattice vertex operator algebra. Let $SU(2)$ be the universal central extension of $SO_3(\mathbb{R})$, let $A:=Z(SU(2))$, and let $G$ be a finite subgroup satisfying

$$A\subseteq G\subseteq SU(2).$$

Write $\overline{G}:=G/A$. Let $\zeta$ be a generator of $H^4(BSU(2),\mathbb{Z})$, let $\omega$ represent the corresponding multiplicative generator of $H^3(G,\mathbb{C}^{\times})$ obtained by restricting $\zeta$ to $G$, and let $D^{\omega}(G,A)$ be the associated modular quasi-Hopf algebra. An $\omega$-admissible pair of $A$ is understood as in the source. The binary polyhedral groups conjecture. For some choice of $\zeta$, there is an equivalence of modular tensor categories

$$V^{\overline{G}}\operatorname{-mod}\simeq \operatorname{Rep}(D^{\omega}(G,A))$$

for some $\omega$-admissible pair of $A$. This proposes a categorical description of the orbifold module category for the binary polyhedral subgroups of $SU(2)$; the supplied text does not indicate whether the conjecture has been proved or disproved.