Local-isolation characterization of binary tropical matrix isoterms

Let $n\geq 2$. For a word $w$ over a two-letter alphabet, call $w$ locally isolated in $\mathcal{UT}_n(\mathbb{T})$ if every word $w'$ satisfying $w'\leftrightarrow w$ is inequivalent to $w$ in $\mathcal{UT}_n(\mathbb{T})$. A word is an isoterm for $\mathcal{UT}_n(\mathbb{T})$ if it forms no nontrivial identity there. Local-isolation conjecture. A word over a two-letter alphabet is an isoterm for $\mathcal{UT}_n(\mathbb{T})$ if and only if it is locally isolated in $\mathcal{UT}_n(\mathbb{T})$.

For $n=2$, this follows from the cited corollary, but the paper conjectures the equivalence for all $n$; it would make local-neighbour checks sufficient to identify all binary isoterms.