Conjectures on ribbon categories of modules for W_Q(r) and B_Q(r)

Let $\mathfrak{g}$ be the Lie algebra with Cartan matrix $A$, rank $\operatorname{rank}(\mathfrak{g})$, root lattice $Q$, weight lattice $P$, dual Cartan $\mathfrak{h}^*$, and Weyl vector $\rho$. Let $r$ be a positive integer, let $a_r=\sqrt{-1/r}$, and let $\operatorname{Rep}_{\langle s\rangle}\mathcal{W}_Q(r)$ and $\operatorname{Rep}_{\langle s\rangle}B_Q(r)$ denote the full subcategories of modules generated by irreducibles.

The conjectures. $\operatorname{Rep}_{\langle s\rangle}\mathcal{W}_Q(r)$ is a finite non-degenerate ribbon category with $\operatorname{det}(A)\cdot r^{\operatorname{rank}(\mathfrak{g})}$ distinct irreducible modules. Moreover, $\operatorname{Rep}_{\langle s\rangle}B_Q(r)$ is non-degenerate and ribbon if $r$ is odd or $\rho\in Q$, and its irreducible modules can be indexed by

$$\{ S^{\mu}_{\gamma}\mid \mu,\gamma\in\mathfrak{h}^*,\ \text{and }\langle\lambda,\mu+ra_r\gamma\rangle\in\mathbb{Z}\text{ for all }\lambda\in P\}$$

with relations $S^{\mu}_{\gamma}\cong S^{\mu+\lambda}_{\gamma+a_r\lambda}$ for all $\lambda\in rP$.

These conjectures extend the rank-one connections between unrolled quantum groups and singlet, triplet, and $B_p$ vertex operator algebras to higher rank. In particular, they predict finite non-degenerate ribbon structures and explicit parametrizations of irreducible modules for the corresponding vertex operator algebras.