Left reversible amenable monoid without maximal global cogrowth

Let $S$ be a finitely generated monoid. Its global cogrowth is maximal when the global cogrowth rate equals the cardinality of a chosen finite generating set. Left amenable means that $S$ admits a left-invariant mean. Existence conjecture. There exists a left reversible finitely generated monoid that is left amenable but does not have maximal global cogrowth.

This asks whether left amenability forces maximal global cogrowth in the monoid setting. The source presents the existence claim as expected and does not report a resolution.