Compact quantum groups generated by their tori

Associated to any closed subgroup $G\subset U_N^{+}$ is a family of toral subgroups $T_Q\subset G$, indexed by the unitary matrices $Q\in U_N$. The family $\{T_Q\mathrm{\mid }Q\in U_N\}$ is expected to encode the main properties of $G$, and there are several conjectures in this sense. We verify here the generation conjecture, $G=<T_Q\mathrm{\mid }Q\in U_N>$, for various classes of compact quantum groups. Our results generalize the previously known facts on the subject.