The finite-set correspondence conjecture for anyonic $SL(N,\mathbb C)$ flat connections

Let $M_3=[\{(p_k,q_k)\}_{k=1}^n]$ be a Seifert manifold, and let $\mathcal R^N_{\{(p_k,q_k)\}_{k=1}^n}$ be the finite set defined by the integer labels satisfying the displayed inequalities and sum condition in the source. Consider the space of anyonic $SL(N,\mathbb C)$ flat connections on $M_3$.

Finite-set correspondence conjecture. The connected components of the anyonic $SL(N,\mathbb C)$ flat-connection space are in one-to-one correspondence with the points of

$$\mathcal R^N_{\{(p_k,q_k)\}_{k=1}^n}.$$

The claim would turn the classification of connected components into a finite combinatorial problem. The authors report numerical evidence for it and defer the numerical investigations to future work.