Nonnegative Demazure-character expansion for Kohnert polynomials of Demazure diagrams

Let $D$ be a demazure diagram, meaning a finite cell diagram satisfying the demazure condition described in the paper, and let ${\mathfrak K}_D$ denote its Kohnert polynomial. The nonnegative expansion conjecture. The polynomial ${\mathfrak K}_D$ expands as a nonnegative sum of Demazure characters. This is motivated by the fact that both composition diagrams and Rothe diagrams are demazure, and by extensive computations supporting the claim.