Maximality Levels of the classical permutation group in the quantum permutation group

Progress on the conjecture of Banica and Bichon that the classical permutation group is a maximal quantum subgroup of the quantum permutation group remains limited to a handful of small-parameter results. By Tannaka–Krein duality, any counterexample to this Maximality Conjecture must arise from a category strictly intermediate between the category $\mathcal{N}\mathcal{C}$ of non-crossing partitions and the category $\mathcal{P}$ of all partitions. Any such exotic category must therefore contain a linear combination of crossing-partition vectors. The categories generated by $\mathcal{N}\mathcal{C}$ together with some such vectors are studied, with a number of generation results. It is shown that no exotic category can contain a linear combination of three crossing-partition vectors, and, at $N=6$, there is no exotic category containing a linear combination of 31 crossing-partition vectors that is distinguished from $\mathcal{N}\mathcal{C}$ or $\mathcal{P}$ at moments of order six.