Decidability of the finiteness problem for automaton groups

Determine whether the finiteness problem for automaton groups is decidable; specifically, given a complete invertible letter-to-letter transducer (a G-automaton), decide whether the generated automaton group is finite.

Background

The finiteness problem for automaton semigroups is known to be undecidable, but the status for automaton groups has resisted resolution.

Establishing decidability or undecidability for automaton groups would close a prominent gap in the algorithmic theory of self-similar and automaton-generated algebraic structures.