Conjecture on the UT-rank of the monoid M_5

Let $M_5 = \operatorname{Mon}\langle a,b\mid aba=ba\rangle$ be the one-relation monoid with defining relation $aba=ba$. Its UT-rank is the least $n$ such that $M_5$ admits a faithful representation into $\operatorname{UT}_n(\overline{\mathbb{Z}})$. UT-rank conjecture for $M_5$. The monoid $M_5$ has UT-rank exactly $4$. This is the remaining question for the UT-tropical embeddings considered here: an embedding into $\operatorname{UT}_4(\overline{\mathbb{Z}})$ is known, but it is not known whether smaller matrices suffice.