Faithfulness and inner faithfulness of the universal flat matrix representation

Let $S_N^+$ be the quantum permutation group, let $X_N$ be the associated matrix-model space, and let

$$\pi_N:C(S_N^+)\to M_N(C(X_N)),\quad \pi_N(w_{ij})=(u\to u_{ij})$$

be the universal $N\times N$ flat matrix representation. Flat matrix representation conjecture. The representation $\pi_N$ is faithful at $N=4$, and is inner faithful at every $N\geq5$. The $N=4$ case is related to proving that the induced scalar functional is Haar integration on $S_4^+$; the $N\geq5$ case is related to convergence of the truncated moments to the Catalan numbers.