Denote by u(n) the largest principal specialization of the Schubert polynomial of a permutation of size n. Stanley conjectured in 2017 that there is a limit $\lim_{n\to \infty} \log u(n)$ and asked for a limiting description of permutations achieving the maximum $u(n)$. Merzon and Smirnov had already conjectured in 2014 that this maximum is achieved on the class of layered permutations. We resolve both Stanley's problems restricted to layered permutations and give improvements on the bound of the possible value for the limit of $\log u(n)$. This is joint work with Igor Pak and Greta Panova.
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