Asymptotic equivalence of tropical matrix identities for random binary words

Fix $r\in(0,1)$ and consider words in $W(\ell r,\ell(1-r))$. Choose $w$ uniformly from this set, choose $u$ uniformly from the $\sim_2$-class of $w$, and choose $v$ uniformly from the immediate neighbours of $w$ in that class. Random-equivalence conjecture. The probabilities

$$\mathbb{P}(w\sim_3 u)\quad\text{and}\quad\mathbb{P}(w\sim_3 v)$$

are monotone increasing in $\ell$, and both tend to $1$ as $\ell\to\infty$.

The conjecture predicts that for long words, most $\mathcal{UT}_2(\mathbb{T})$ identities, especially those obtained by immediate adjacent swaps, also hold in $\mathcal{UT}_3(\mathbb{T})$.