Tableaux problem 13

13.[MerSmir2015a]*Conj. 5.7 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, prove that the permutation $w\in S_{n}$ that maximizes $\nu_{w}$ is “Richardson” or “layered” in that it has the form [Def. 5.5] $(i_{1},i_{1}-1,\ldots,2,1,i_{2},i_{2}-1,\ldots,i_{1}+1,i_{3},\ldots,i_{2}+1,\ldots)$ for some $i_{1}<i_{2}<i_{3}<\cdots<i_{k}=n$.