  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.
