Maximality conjecture for the symmetric group in the quantum permutation group

Let $\mathcal Q_n$ be the quantum permutation group on $n$ points, containing the classical symmetric group $S_n$. Maximality conjecture. The symmetric group $S_n$ is maximal in the quantum permutation group $\mathcal Q_n$, for any $n$. This conjecture concerns the classification of quantum subgroups of $\mathcal Q_n$ and asks whether the classical symmetric group can be enlarged within $\mathcal Q_n$ only trivially. The surrounding discussion indicates that the classification problem is understood as difficult for general $n$ and is expected to be accessible first for small values such as $n=5,6,7$.