Adjacent-swap connectivity conjecture for binary tropical matrix identities

Let $\Sigma$ be a two-letter alphabet, let $w,v\in\Sigma^+$ with $w\neq v$, and let $n\geq 2$. Write $w\leftrightarrow v$ when $w$ and $v$ differ by one adjacent swap, and let $\sim_n$ denote equivalence in $\mathcal{UT}_n(\mathbb{T})$. Adjacent-swap connectivity conjecture.

$$w\sim_n v$$

if and only if there exists a positive integer $k$ and words $u(0),\ldots,u(k)$ such that

$$w=u(0)\leftrightarrow u(1)\leftrightarrow\cdots\leftrightarrow u(k)=v$$

and

$$u(0)\sim_n u(1)\sim_n\cdots\sim_n u(k).$$

The conjecture is asserted only for binary alphabets; the paper notes that it fails for alphabets of size greater than two, while it holds trivially for the previously known identities arising from a single adjacent swap.