Pattern-containment characterization of SCNP for dual Schubert polynomials

Let $S_n$ be the symmetric group on $n$ letters. For permutations $u,w\in S_n$, let $D_u^w$ denote the dual Schubert polynomial, and let SCNP denote the property that its Newton polytope is a generalized permutahedron. SCNP pattern-containment conjecture. For a permutation $u\in S_n$, there exists a permutation $w\in S_n$ such that $D_u^w$ does not have SCNP if and only if $u$ contains the pattern $1324$. Analogously, for a permutation $w\in S_n$, there exists a permutation $u\in S_n$ such that $D_u^w$ does not have SCNP if and only if $w$ contains the pattern $4231$. The conjecture gives a proposed pattern-theoretic characterization of when dual Schubert polynomials have SCNP; it has been verified computationally for $S_n$ with $n\leq 6$, while the general case remains open.