Wang's finiteness conjecture for modular categories of fixed rank

A modular category is a braided fusion category equipped with a compatible ribbon structure whose associated braiding is non-degenerate; its rank $n$ is the number of isomorphism classes of simple objects.

Wang's finiteness conjecture. There are finitely many modular categories of rank $n$ for any fixed $n\in\mathbb{N}$.

This conjecture concerns the classification of modular categories after fixing their rank. The surrounding discussion notes that Ocneanu rigidity gives finiteness for a fixed fusion algebra, but it remains open whether only finitely many modular categories can occur at each fixed rank.