Finiteness conjecture for Woronowicz algebras with a fixed fusion-list invariant

Let $A$ be a Woronowicz algebra, and let $(R^+(A),list)$ denote its fusion semiring together with the invariant assigning to each corepresentation the list of positive numbers determined by its associated matrix. A given pair $(R^+,list)$ is fixed.

Finiteness conjecture. There are only finitely many Woronowicz algebras having a given $(R^+,list)$ invariant.

The invariant $(R^+,list)$ is stronger than the dimension and quantum-dimension invariants and is known to classify several families, but the asserted finiteness for every fixed invariant is not established here.