Banica–Vergnioux refined freeness conjecture for easy quantum groups

Let $\mathbb{G}$ be a free unitary easy quantum group without any non-trivial one-dimensional representation. Let $\mathcal{C}^{\circ,\bullet}$ be its category of noncrossing colored partitions, and let $S(\mathcal{C}^{\circ,\bullet})$ be the set of equivalence classes of one-block projective partitions. The fusion semiring of $\mathbb{G}$ is $(R^{+}(\mathbb{G}),-,\otimes)$. Banica–Vergnioux's refined freeness conjecture. There is an isomorphism of fusion semirings

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

This formulation removes the obstruction caused by non-trivial one-dimensional representations and is supported by explicit computations for several free easy quantum groups. Whether it holds in the stated generality remains open.