Liouville property conjecture for degree-2 mother groups and automaton groups

Let an automaton group have degree $2$, meaning that every element $g$ has activity $a_n(g)=O(n^2)$, and let the associated degree-$2$ mother groups be the specific universal automaton groups used to contain degree-$2$ automaton groups as subgroups. A random walk is understood in the usual sense on such a group, and a group is Liouville for a random walk when every bounded harmonic function is constant. Liouville property conjecture. The mother groups of degree $2$ are Liouville with respect to some, or even every, random walk on them. Moreover, the same holds for every automaton group of degree $2$. This asks whether the Liouville method proving amenability in degrees $0$ and $1$ extends to degree $2$; the paper's main theorem establishes amenability for degree-$2$ automaton groups, while this stronger Liouville assertion is posed as a conjecture.