Decay of the isoterm fraction in binary tropical matrix identities

Let $W(\ell r,\ell(1-r))$ denote the set of binary words with the indicated numbers of the two letters. Fix $r\in(0,1)$ and $n\geq 2$, and let $i(n,\ell,r)$ be the fraction of $\mathcal{UT}_n(\mathbb{T})$ isoterms in $W(\ell r,\ell(1-r))$. Isoterm-fraction conjecture. The function $i(n,\ell,r)$ is monotone decreasing in $\ell$, and

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

This formalizes the experiments suggesting that, at a fixed letter ratio, isoterms become asymptotically rare among long binary words.