Adamović–Wang duality conjecture for affine and Virasoro vertex algebras

Let $q\geq 2$ with $3\nmid q$, and define the generalized vertex operator algebras

$$A^{(q)}=L_{-2+q}(\mathfrak{sl}_2)\oplus L_{-2+q}(\mu_{q-2,0}),$$ $$B^{(q)}=L_\mathrm{Vir}(c_{q,3},0)\oplus L_\mathrm{Vir}(c_{q,3},h_{q-1,1}^{q,3}).$$

Duality means that each simple vertex (super)algebra can be equipped with a module structure over the other, isomorphic to the other algebra, with each algebra realized as a subalgebra of the vertex algebra of local fields acting on modules over the other. Adamović–Wang duality conjecture. The vertex operator algebras $L_{-2+q}(\mathfrak{sl}_2)$ and $L_\mathrm{Vir}(c_{q,3},0)$ are dual in this sense; similarly, $A^{(q)}$ and $B^{(q)}$ are dual. In particular, $L_{-2+q}(\mathfrak{sl}_2)$ can be equipped with the structure of an irreducible $L_\mathrm{Vir}(c_{q,3},0)$-module isomorphic to $L_\mathrm{Vir}(c_{q,3},0)$, and vice versa. The case $q=4$ is established in the paper, while the extension to general $q$ is posed as a possibility and remains open.