Banica's universal flat matrix model conjecture for quantum permutation groups

Let $S_N^+$ be the quantum permutation group, and let $X_N$ be the compact space of $N\times N$ matrices $P=(P_{ij})$ whose entries are rank-one projections in $M_N(\mathbb C)$ satisfying

$$\sum_jP_{ij}=1=\sum_iP_{ij}.$$

The universal flat matrix model is the representation

$$\pi:\mathcal A(S_N^+)\to M_N(C(X_N)),\qquad \pi(u_{ij})=\{P\mapsto P_{ij}\}.$$

A representation is faithful when it is injective, and inner faithful when its kernel contains no non-trivial Hopf $\ast$-ideals. Universal flat matrix model conjecture. The universal flat matrix model is faithful for $N=4$ and inner faithful for $N\geq5$. The conjecture is attributed to Banica; the source does not provide evidence of resolution.