Modularity conjecture for admissible affine Lie algebra modules

Let $\mathfrak g$ be a simple Lie algebra, with dual Coxeter number $h^\vee$ and lacety $d$. Let $\ell=-h^\vee+\frac{a}{b}$ be an admissible level, where $a,b$ are coprime positive integers, and set

$$q=e^{\frac{2\pi i}{2d(\ell+h^\vee)}}.$$

Let $\mathcal O_{\ell,\operatorname{ord}}$ be the braided tensor category constructed in the paper. A braided tensor category is modular when it has the required nondegenerate modular tensor-category structure. Modularity conjecture. The category $\mathcal O_{\ell,\operatorname{ord}}$ is modular except in the following cases, with $n\in\mathbb Z_{>0}$:

1. $\mathfrak g\in\{\mathfrak{sl}_{2n},\mathfrak{so}_{2n},\mathfrak e_7,\mathfrak{sp}_n\}$ and $b$ is even.
2. $\mathfrak g=\mathfrak{so}_{4n+1}$ and $b\equiv0\pmod 4$.
3. $\mathfrak g=\mathfrak{so}_{4n+3}$ and $b\equiv2\pmod 4$.

This conjecture concerns the nondegeneracy of the braided tensor categories arising from admissible affine Lie algebra modules. The paper proves it for $\mathfrak g=\mathfrak{sl}_2$, while the general assertion remains open in the supplied text.