Freeness conjecture for fusion semirings of free quantum algebras

Let $A$ be a free quantum algebra, meaning a Hopf algebra satisfying

$$A_u(n)\to A\to A_s(n)$$

whose tensor category is spanned by partitions. Let $R^+(A)$ denote its fusion semiring. Freeness conjecture. If $A$ is free, then $R^+(A)$ is free.

The fusion rules of the known free quantum algebras are described by free fusion semirings, and the conjecture seeks to extend this pattern to all free quantum algebras. A satisfactory axiomatization of the underlying involution and fusion data is not known.