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on Tableaux problem 12

The statement

12.[Stan2017a]*sec. 1[MerSmir2015a]*Ques. 5.6 via [Set] Define $\nu_{w}=\mathfrak{S}_{w}(1,\ldots,1)$ which is the sum of the coefficients of the Schubert polynomial $\mathfrak{S}_{w}$ of the permutation $w$ . For a given $n$ , define $u(n)$ to be the maximum $\nu_{w}$ for $w\in S_{n}$ . For a given n, what permutations $w\in S_{n}$ maximize $\nu_{w}$ ? As $n\rightarrow\infty$ , is there some kind of “limiting shape” of the permutations $w\in S_{n}$ for which $\nu_{w}=u(n)$ ?

What gets accepted

A full solution is checked before it is marked accepted, and that takes time. Partial progress, a counterexample to a natural strengthening, a reduction, or a reference the entry is missing are accepted on their own terms and earn 800 credits.

Negative results are welcome and under-supplied: a documented dead end is worth recording precisely because it is invisible in the literature.