Conjecture on tropical representations of the monoid M_4

Let $M_4 = \operatorname{Mon}\langle a,b\mid aba^2=ba\rangle$ and let $M_7$ denote its anti-isomorphic one-relation monoid. A faithful tropical representation of a monoid is an injective homomorphism into $\operatorname{M}_n(\overline{\mathbb{Z}})$ for some finite $n$. Tropical non-representability conjecture for $M_4$. The monoid $M_4$ does not admit a faithful tropical representation into $\operatorname{M}_n(\overline{\mathbb{Z}})$ for any $n\geq 1$; hence, neither does $M_7$. The paper identifies $M_4$ and $M_7$ as open cases and records the author's belief that neither is M-tropical, but no proof is given.