SNP conjecture for double Schubert polynomials

Let $X=\{x_1,x_2,\ldots\}$ and $Y=\{y_1,y_2,\ldots\}$, and let ${\mathfrak S}_w(X;Y)$ be the double Schubert polynomial defined recursively from the product for the longest permutation. A polynomial has SNP when every lattice point of its Newton polytope is an exponent vector. Double Schubert-polynomial SNP conjecture.

$${\mathfrak S}_w(X;Y)\text{ is SNP}.$$

The claim was checked for $n\leq5$ and many other cases, and it implies the ordinary Schubert-polynomial conjecture by setting $Y=0$, but remains open generally.