Identity conjecture for tropical matrix monoids

For each positive integer $n$, let $M_n(\mathbb T)$ be the monoid of $n\times n$ tropical matrices. A nontrivial semigroup identity is an identity between distinct semigroup words that holds under every substitution of elements of the monoid. Identity conjecture. The monoid $M_n(\mathbb T)$ satisfies a nontrivial semigroup identity for all $n$. This conjecture was known to hold for $n=2$ by the result cited in the source, while the general case is left open there.