Finite-state weakly branch groups with non-abelian free quotients

Determine whether there exists a weakly branch group generated by finitely many finite-state elements that has a non-abelian free quotient.

Background

The paper constructs weakly branch groups with free quotients by embedding subgroups of the automorphism group of a rooted tree through monomorphisms whose images are generally not represented by finite-state automata. The author observes that any finite-state element in the relevant embedded image must have finite order, so the construction does not produce nontrivial finite-state elements in the embedded free subgroup.

The unresolved question asks whether the existence of a non-abelian free quotient can nevertheless be achieved by a weakly branch group generated by finitely many finite-state elements. This would provide a finite-state analogue of the paper’s main construction and clarify how restrictive finite-state generation is for quotients of weakly branch groups.