The intermediate-word length conjecture for K-Knuth equivalent tableaux

Let $T$ and $T'$ be tableaux with $T\equiv T'$. For a tableau $T$, let its size be the number of boxes in its shape, and let $w\overset{k}{\equiv}w'$ mean that $w$ and $w'$ can be connected by $K$-Knuth moves through words of length at most $k$. Intermediate-word length conjecture. If $k$ is the largest size of a tableau $K$-Knuth equivalent to $T$ or $T'$, then

$$\mathfrak{row}(T)\overset{k}{\equiv}\mathfrak{row}(T').$$

The theorem proved immediately beforehand gives a weaker polynomial upper bound, while this conjecture asserts that the maximum tableau size in the equivalence class suffices. Its status is not resolved in the supplied text.