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on Tableaux problem 14

The statement

14.[MoraPakPan2019a]*Conj. 4.1 via [Set] Define $\Upsilon_{w}=\mathfrak{S}_{w}(1,\ldots,1)$ which is the sum of the coefficients of the Schubert polynomial $\mathfrak{S}_{w}$ of the permutation $w$ . For permutations $w$ and $v$ , define $w\otimes v$ to be the permutation whose permutation matrix is the Kronecker product of the permutation matrices of $w$ and $v$ . Define $1^{n}$ as the identity permutation in $S_{n}$ . Is it true that for any integer $k\geq 2$ , $\Upsilon_{w\otimes 1^{k}}\geq\Upsilon_{w}^{k^{2}}$ ?

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.