Edelman–Greene maximality and height conjecture

Let $Q\in\mathrm{SYT}(\mathrm{sc}_n)$, let $c(Q)$ be its column word, let $\mathcal{P}_Q$ be the associated poset, and let $\mathrm{EG}^{-1}(Q)$ be its inverse Edelman–Greene image. Let $h(P)$ denote the length of the longest chain in a poset $P$, and define

$$\ell_Q=h\left([c(Q),\mathrm{EG}^{-1}(Q)]\right).$$

Edelman–Greene height conjecture. The word $\mathrm{EG}^{-1}(Q)$ is a maximal element of $\mathcal{P}_Q$, and

$$\ell_Q=\sum_{i=1}^{\operatorname{len}(c(Q))}\left(\mathrm{EG}^{-1}(Q)_i-c(Q)_i\right).$$

The first assertion restates the preceding maximality conjecture, while the equality is motivated by computations for $Q\in\mathrm{SYT}(\mathrm{sc}_n)$. The source notes that the maximal element is not generally unique and gives no evidence that the height equality has been proved.