Finiteness conjecture for orbit groups of finite automaton groups

Let $G$ be a finite automaton group, and let $G_{\mathcal O}$ denote the orbit group associated with an orbit $\mathcal O$ of $G$. Orbit-group finiteness conjecture. For each finite automaton group $G$, all of the orbit groups $G_{\mathcal O}$ are finite. The conjecture would imply that the orbit-automaton construction cannot produce an infinite orbit group from a finite automaton group; the source notes that no counterexamples are known.