Decay of the isoterm-class ratio for binary tropical matrix identities

Fix $r\in(0,1)$. For binary words in $W(\ell r,\ell(1-r))$, let $e(\ell,r)$ be the ratio of the number of $\mathcal{UT}_2(\mathbb{T})$ isoterms to the number of $\mathcal{UT}_2(\mathbb{T})$ equivalence classes. Isoterm-class ratio conjecture. The function $e(\ell,r)$ is monotone non-increasing in $\ell$, and

$$e(\ell,r)\to 0\quad\text{as }\ell\to\infty.$$

This conjecture expresses the experimental observation that, at a fixed letter ratio, isoterm classes become asymptotically negligible among equivalence classes of long words.