Kronecker-product supermultiplicativity conjecture for principal Schubert evaluations

For $w\in S_n$, let $w\otimes 1^c$ be the Kronecker product permutation of size $cn$, whose permutation matrix is the Kronecker product of the permutation matrix $P_w$ and the identity matrix $I_c$. Let $\Upsilon_w=\mathfrak{S}_w(1,\ldots,1)$.

Kronecker-product supermultiplicativity conjecture. For every $w\in S_n$,

$$\Upsilon_{w\otimes 1^2}\geq \Upsilon_w^4.$$

The conjecture was verified computationally for all $w\in S_n$ with $n\leq5$, but remains open.