Polynomial-growth obstruction to amenability for left reversible semigroups

Let $S$ be a left reversible semigroup if any two principal left ideals intersect, and call $S$ finitely generated when a finite subset generates it. Its growth is polynomial when its growth function is bounded above by a polynomial. Polynomial-growth obstruction conjecture. There is a left reversible, finitely generated semigroup of polynomial growth which is not left amenable. The source presents this as an obstruction to the possibility that every left reversible, finitely generated semigroup of subexponential growth is left amenable; it remains open in the supplied text.