The Edelman–Greene coefficient bound

Let $w$ be a permutation and let $a_{w,\lambda}$ denote the corresponding Edelman–Greene coefficient for a partition $\lambda$, with $f^{\lambda}$ the number of standard Young tableaux of shape $\lambda$. Edelman–Greene coefficient bound. One has

$$a_{w,\lambda}\leq f^{\lambda}.$$

This bound was stated in the paper as a conjecture and has since been proved by G. Orelowitz in private communication.