Generalised Catalan conjecture for largest binary tropical matrix equivalence classes

Let $W(\ell_a,\ell_a)$ be the set of binary words containing $\ell_a$ occurrences of each letter, and consider its $\mathcal{UT}_2(\mathbb{T})$ equivalence classes. Generalised Catalan conjecture. For each fixed $\ell_a\geq 5$, the largest equivalence class in $W(\ell_a,\ell_a)$ belongs to the generalised Catalan family defined in the source.

The conjecture concerns the extremal structure of equivalence classes with balanced letter content. It has been verified numerically by exhaustive enumeration through words of length $24$, but no general proof is supplied.