The strengthened row-maximum conjecture for plactic centralizers

Let $u$ be an element of the plactic monoid, let $m=\max u$, and let $\ell$ be the number of rows of $P(u)$. For $w\in C(u)$, write the rows of $P(w)$ as $R_i$ for $i\geq 1$.

Strengthened row-maximum conjecture. For every $i$ with $1\leq i\leq \ell$,

$$\max R_i\leq m.$$

This would strengthen the row-maximum bound used in the paper to characterize centralizers in the plactic monoid, and is presented as a direction for future research; its resolution is not established here.