Doubly layered maximizer conjecture at the principal specialization q = -1

Let $S_n$ be the symmetric group, let $\Phi_w$ denote the principal specialization of the Schubert polynomial at $q=-1$, and let $\mathcal{DL}_n$ denote the set of doubly layered permutations. Define

$$\widetilde{u}(n):=\max_{w\in S_n}\Phi_w,\qquad \widetilde{v}(n):=\max_{w\in\mathcal{DL}_n}\Phi_w.$$

Doubly layered maximizer conjecture. For every $n$, all permutations reaching the maximum $\widetilde{u}(n)$ are doubly layered permutations. Thus, there is a limit

$$\lim_{n\to\infty}\frac{\log_2\widetilde{u}(n)}{n^2}.$$

The paper proves the corresponding asymptotic formula for the doubly layered maximum $\widetilde{v}(n)$, but the unrestricted maximization and the asserted limit remain open.