Uniqueness of 2-full words modulo plane symmetries and hook-snake flips

Let $h>w\geq 5$ with $h\equiv_3 w$, and let $W\in\mathcal{MW}^{\leq 2}_{h\times w}$ be a 2-full word, meaning a maximal 2-dimensional binary word of degree at most $2$ in an $h\times w$ rectangle. The allowed equivalences are symmetries of the plane together with a local symmetry, namely a flip, on hook snakes.

Uniqueness conjecture for 2-full words. The 2-full words $W\in\mathcal{MW}^{\leq 2}_{h\times w}$ are unique up to symmetries of the plane and a local symmetry (a flip) on hook snakes.

The question concerns the exact structure of maximal bounded-degree binary words. The supplied status evidence states that this uniqueness question is open; the paper's main theorem gives bounds for maximal snakes but does not settle their exact length in terms of $h$ and $w$.