Tableaux problem 14

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 $k\geq 2$, $\Upsilon_{w\otimes 1^{k}}\geq\Upsilon_{w}^{k^{2}}$?