Inner faithfulness of the universal quasi-flat model under uniformity

Let $G\subset S_N^+$ be quasi-transitive, with all orbits of size $K$, and let

$$\pi:C(G)\to M_K(C(X_G))$$

be its universal quasi-flat model. The uniformity conjecture. If $G$ additionally satisfies a suitable uniformity condition, then the universal quasi-flat model is inner faithful. The paper presents this as a conjectural approach to an old open problem concerning inner faithful matrix models for quantum permutation groups; the suitable uniformity condition is not specified, and no proof is given.