Sums of Schubert structure constants with bounded Coxeter length

Pak-Robichaux recently introduced a signed puzzle rule for Schubert structure constants, which they use to show that sums $\gamma_k(n)$ of these constants with a bounded number of inversions are polynomial. We give a different, conceptual proof of their theorem. Our argument computes the lead term of $\gamma_k(n)$ and extends to all classical Lie types.