A Concrete Model for the Quantum Permutation Group on 4 Points

In 2019, Jung-Weber gave an example of a concrete magic unitary $M$, which defines a $C^{\ast }$-algebraic model of the quantum permutation group $S_4^{+}$. We show with the help of a computer that there exist no polynomials up to degree $50$ separating the entries of $M$ from the generators of $C(S_4^{+})$. This indicates that the magic unitary $M$ might already define a faithful model of $S_4^{+}$.