Dichotomy conjecture for free products of automaton semigroups

Let $\mathcal{A}$ and $\mathcal{B}$ be automata, and let $\Sigma(\mathcal{A})$ and $\Sigma(\mathcal{B})$ be the automaton semigroups generated by their states. Their free product is denoted by $\Sigma(\mathcal{A})\star\Sigma(\mathcal{B})$. Dichotomy conjecture. One of the following holds: every free product of two, and hence of finitely many, automaton semigroups is itself an automaton semigroup; or it is undecidable, given two automata $\mathcal{A}$ and $\mathcal{B}$, whether $\Sigma(\mathcal{A})\star\Sigma(\mathcal{B})$ is an automaton semigroup. This presents the unresolved closure question as a dichotomy between universal closure and undecidability of the corresponding decision problem.