SNP conjecture for Demazure atoms

Let $\alpha$ be a weak composition, and let $A_\alpha$ be the Demazure atom obtained from $x^\alpha$ by applying the operators $\widehat\pi_i=\pi_i-\operatorname{id}$ according to the recursive definition in the paper. A polynomial has SNP when every lattice point of its Newton polytope is an exponent vector. Demazure-atom SNP conjecture. The Demazure atom $A_\alpha$ has SNP. The source presents this as an open conjecture.