Stanley’s asymptotic maximum for principal specializations

Determine whether the limit L = lim_{n→∞} (1/n^2) log_2 (max_{w∈S_n} 𝛶_w) exists for Schubert principal specializations 𝛶_w = 𝔖_w(1^n); if it exists, compute its value and identify the permutations w that achieve the maximum for large n.

Background

Schubert polynomials’ principal specializations 𝛶_w equal the number of reduced (bumpless) pipe dreams with boundary permutation w. Stanley asked for the leading n^2-scale asymptotics of the maximum of these specializations over S_n.

Lower and upper bounds are known, and layered permutations provide an explicit lower bound construction, but recent computations here disprove the conjecture that layered permutations always maximize 𝛶_w. Despite progress and computational evidence, the fundamental asymptotic question remains unresolved.