Simplicity and unique trace conjecture for quantum groups satisfying freeness

Let $\mathbb{G}$ be a compact quantum group, and suppose that its fusion semiring satisfies the freeness conjecture. Simplicity and unique trace conjecture. The reduced $C^*$-algebra of $\mathbb{G}$ is simple and has a unique trace.

This conjecture connects free fusion rules with operator-algebraic rigidity. The source presents it as a companion conjecture; its status is not resolved there.