SNP conjecture for Schubert polynomials

For $w\in S_\infty$, let ${\mathfrak S}_w$ be the Schubert polynomial defined recursively from the longest permutation using divided-difference operators. A polynomial has SNP when every lattice point of its Newton polytope is an exponent vector. Schubert-polynomial SNP conjecture.

$${\mathfrak S}_w\text{ has SNP}.$$

The conjecture was checked for all $w\in S_n$ with $n\leq8$, but is open in general in the source.