Principal specializations of Schubert polynomials, multi-layered permutations and asymptotics

Let $v\left(n\right)$ be the largest principal specialization of Schubert polynomials for layered permutations $v\left(n\right):={\max }_{w\in L_n}{S}_w(1,\dots ,1)$. Morales, Pak and Panova proved that there is a limit$\underset{n\to \infty }{\lim }\frac{\log v\left(n\right)}{n^2},$ and gave a precise description of layered permutations reaching the maximum. In this paper, we extend Morales Pak and Panova's results to generalized principal specialization $S_w(1,q,q^2,\dots )$ for multi-layered permutations when q equals a root of unity.