Homogeneous quantum groups and their easiness level

Given a closed subgroup $G\subset U_N^{+}$ which is homogeneous, in the sense that we have $S_N\subset G\subset U_N^{+}$, the corresponding Tannakian category $C$ must satisfy $span({\mathcal{N}\mathcal{C}}_2)\subset C\subset span(P)$. Based on this observation, we construct a certain integer $p\in \mathbb{N}\cup \{\mathrm{\infty }\}$, that we call “easiness level” of $G$. The value $p=1$ corresponds to the case where $G$ is easy, and we explore here, with some theory and examples, the case $p>1$. As a main application, we show that $S_N\subset S_N^{+}$ and other liberation inclusions, known to be maximal in the easy setting, remain maximal at the easiness level $p=2$ as well.