The homological modularity conjecture for relative fusion categories

Fix $N\geq 2$. Let $\{(p_k,q_k)_{k=1}^n\}$ be Seifert data with $p_k\geq N$ and $\gcd(N,q_k)=1$, and assume that at most one $p_k$ equals $N$. Set

$$M_3=[\{(p_k,q_k)\}_{k=1}^n].$$

The associated relative fusion category is $\mathsf{RFC}(\mathfrak{sl}(N),M_3)$.

Homological modularity conjecture. The category $\mathsf{RFC}(\mathfrak{sl}(N),M_3)$ is modular if and only if

$$H_1(M_3;\mathbb Z_N)=0.$$

This conjecture identifies non-degeneracy of the category's $S$-matrix with vanishing first homology with $\mathbb Z_N$ coefficients. It is the second main conjecture of the paper and remains open in the stated generality.