Finite counterexample to free-product closure for automaton semigroups

Let $S$ and $T$ be finite semigroups, and let $S\star T$ denote their semigroup free product. An automaton semigroup is a semigroup recognized by a finite automaton acting on finite words. Finite free-product counterexample conjecture. There exist finite semigroups $S$ and $T$ such that $S\star T$ is not an automaton semigroup. The paper suggests that this might be proved using very small examples, such as the trivial semigroup and a two-element null semigroup, but no proof is given.