The characters conjecture for connected compact quantum groups

Let $G\subset U_N^+$ be a connected compact quantum group. Its central algebra $C(G)_{central}$ is the norm closure of the linear span of irreducible representation characters. For each $Q\in U_N$, let $\pi_Q:C(G)\to C^*(\Gamma_Q)$ be the associated quotient map. Characters conjecture. For every nonzero $P\in C(G)_{central}$, there exists $Q\in U_N$ such that

$$\pi_Q(P)\neq 0.$$

The claim is trivial for group duals and holds for classical groups, while the stronger possibility that one such $\pi_Q$ is faithful on $C(G)_{central}$ remains part of the proposed picture.