Strict identity-variety growth conjecture for upper triangular tropical matrices

Let $b$\mathbb{T}$be the tropical semiring, and let$UT_n(b$\mathbb{T})$ denote the semigroup of $n\times n$ upper triangular tropical matrices. A semigroup identity is a formal equality of words that holds under every substitution of matrices.

Strict identity-variety growth conjecture. For every positive integer $n$, there is a semigroup identity satisfied by $UT_n(b$\mathbb{T})$but not by$UT_{n+1}(b$\mathbb{T})$.

The paper proves that no single semigroup identity holds for all finite-dimensional upper triangular tropical matrix semigroups, but the stronger assertion that each successive dimension has strictly fewer identities is left as an expectation.