Asymptotics of principal evaluations of Schubert polynomials for layered permutations

Denote by $u(n)$ the largest principal specialization of the Schubert polynomial: $u(n):={\max }_{w\in S_n}{\mathfrak{S}}_w(1,\dots ,1)$ Stanley conjectured in [arXiv:1704.00851] that there is a limit $\lim_{n\to \infty} , \frac{1}{n^2} \log u(n), $ and asked for a limiting description of permutations achieving the maximum $u(n)$. Merzon and Smirnov conjectured in [arXiv:1410.6857] that this maximum is achieved on layered permutations. We resolve both Stanley’s problems restricted to layered permutations.