Banica–Vergnioux freeness conjecture for free unitary easy quantum groups

Let $\mathbb{G}$ be a free unitary easy quantum group. A free fusion semiring is obtained from a set $S$ with an involution and a binary law by taking the free monoid $R(S)$, with the induced involution and tensor product. The fusion semiring of $\mathbb{G}$ is denoted $(R^{+}(\mathbb{G}),-,\otimes)$. Banica–Vergnioux's freeness conjecture. There is a set $S$ together with an involution and a binary law such that

$$(R^{+}(\mathbb{G}),-,\otimes)\simeq (R^{+}(S),-,\otimes).$$

The conjecture proposes that the fusion rules of free unitary easy quantum groups have a universal free structure. In the stated form it is false, for example for the symmetrized quantum permutation group $S_N^{+}\times\mathbb{Z}_2$; additional assumptions are needed, and the precise corrected formulation remains open.