SNP conjecture for key polynomials

Let $\alpha=(\alpha_1,\alpha_2,\ldots)\in\mathbb Z_{\geq0}^{\infty}$ have finite size, and let $\kappa_\alpha$ be the key polynomial defined using Demazure operators. A polynomial has SNP when every lattice point of its Newton polytope is represented by an exponent vector. Key-polynomial SNP conjecture. The key polynomial $\kappa_\alpha$ has SNP. The claim is presented as conjectural; the paper gives related partial results but no general resolution.