The freeness conjecture for two-state reversible Mealy automata

Let a reversible Mealy automaton be a Mealy automaton whose transition maps are permutations, and let the semigroup generated by an automaton be the semigroup generated by its states under composition. Two-state freeness conjecture. Every $2$-state reversible Mealy automaton generates a semigroup which is either finite or free of rank $2$. This is a structural conjecture about infinite automaton semigroups. The source reports that it was tested and appeared correct for reversible $2$-state Mealy automata with up to $6$ letters.