The amenability conjecture for compact quantum groups

Let $G\subset U_N^+$ be a compact quantum group. Write $C_{\max}(G)$ and $C_{\min}(G)$ for its maximal and minimal Hopf $C^*$-algebras; $G$ is coamenable when the canonical quotient $C_{\max}(G)\to C_{\min}(G)$ is an isomorphism. For each $Q\in U_N$, let $\Gamma_Q$ be the associated compact quantum group. Amenability conjecture. The quantum group $G$ is coamenable if and only if every $\Gamma_Q$ is coamenable.

The forward implication is immediate from the quotient maps $C(G)\to C^*(\Gamma_Q)$. The reverse implication is the substantive part and is trivial for group duals and classical compact groups; it remains open in general.