Invariable generation of certain branch groups

Let $G$ be a group. Then $S\subseteq G$ is an invariable generating set of $G$ if every subset $S^{\mathrm{\prime }}$ obtained from $S$ by replacing each element with a conjugate is also a generating set of $G$. We investigate invariable generation among key examples of branch groups. In particular, we prove that all generating sets of the torsion Grigorchuk groups, of the branch Grigorchuk-Gupta-Sidki groups and of the torsion multi-EGS groups (which are natural generalisations of the Grigorchuk-Gupta-Sidki groups) are invariable generating sets. Furthermore, for the first Grigorchuk group and the torsion Grigorchuk-Gupta-Sidki groups, every finitely generated subgroup has a finite invariable generating set. Our results apply to finitely generated groups in $\mathcal{M}\mathcal{N}$, the class of groups whose maximal subgroups are all normal. We then obtain that any $2$-generated group in $\mathcal{M}\mathcal{N}$ is almost $\frac{3}{2}$-generated, and end by applying this observation to generating graphs.