Nonnegative Schubert expansion conjecture for skew polynomials

Let $a$ be a weak composition, let $\mathbb S(a)$ be the associated skew diagram, and let ${\mathfrak K}_{\mathbb S(a)}$ be its skew polynomial. The skew-polynomial expansion conjecture. The skew polynomial expands as a nonnegative sum of Schubert polynomials. This places Schubert polynomials between Demazure characters and skew polynomials in the proposed hierarchy of polynomial bases. The status evidence supplied with the source indicates that this conjecture is disproved, so the asserted nonnegative expansion does not hold in general.