  Given a quantum permutation group $G\subset S_N^+$, with orbits having the
same size $K$, we construct a universal matrix model $\pi:C(G)\to M_K(C(X))$,
having the property that the images of the standard coordinates $u_{ij}\in
C(G)$ are projections of rank $\leq 1$. Our conjecture is that this model is
inner faithful under suitable algebraic assumptions, and is in addition
stationary under suitable analytic assumptions. We prove this conjecture for
the classical groups, and for several key families of group duals.
