Maximizing the Edelman-Greene statistic

The $\textit{Edelman-Greene statistic}$ of S. Billey-B. Pawlowski measures the "shortness" of the Schur expansion of a Stanley symmetric function. We show that the maximum value of this statistic on permutations of Coxeter length $n$ is the number of involutions in the symmetric group $S_n$, and explicitly describe the permutations that attain this maximum. Our proof confirms a recent conjecture of C. Monical, B. Pankow, and A. Yong: we give an explicit combinatorial injection between a certain collections of Edelman-Greene tableaux and standard Young tableaux.