Characterization of the tableaux-graph order by row entries

Let $S,T\in\mathcal{T}(n,n)$ be standard Young tableaux of shape $(n,n)$, and let $a_i$ and $b_i$ denote the entries in the first row of $S$ and $T$, respectively. Characterization conjecture. If $a_i\geq b_i$ for all $1\leq i\leq n$, then $S\leq T$ in the tableaux graph. Together with the preceding implication proved in the paper, this would characterize the tableaux-graph order by coordinatewise comparison of first-row entries. The supplied text presents this as part of the proof of the main conjecture and gives no resolution status beyond that presentation.