The elementary amenability conjecture for generalized basilica groups

Let $X$ be a finite set, let $X^*$ be the rooted tree given by the free monoid on $X$, and let $G\leq\operatorname{Aut}(X^*)$ be a generalized basilica group: namely, $G$ admits a finite generating set $Y$ such that for every $g\in Y$ either $g_x=1$ for all $x\in X$, or $g_x\in\{1,g\}$ for all $x\in X$, with exactly one $x$ such that $g_x=g$. Here $g_x$ denotes the section of $g$ at $x$. The elementary amenability conjecture for generalized basilica groups. Every generalized basilica group is either (locally finite)-by-(virtually abelian) or not elementary amenable. The theorem in the paper proves this dichotomy for balanced generalized basilica groups, while the conjecture extends it to all generalized basilica groups.