Classification conjecture for right cyclic pivotal unitary Temperley–Lieb–Jones modules

Let $\Gamma$ be the fixed data defining the unitary $*$-2-category $\operatorname{TLJ}(\Gamma)$. A MW-type bipartite balanced $\Gamma$-fair graph is a bipartite balanced $\Gamma$-fair graph satisfying the MW condition described in the surrounding classification framework. Classification conjecture. Equivalence classes of right cyclic pivotal unitary $\operatorname{TLJ}(\Gamma)$-modules correspond to MW-type bipartite balanced $\Gamma$-fair graphs. This conjectures a generalization to the $*$-2-categorical setting of the classification of right cyclic pivotal Temperley–Lieb–Jones $C^*$-modules by bipartite graphs equipped with a dimension function satisfying a Perron–Frobenius condition. The status of this generalization is left open for future work.