Nonnegativity conjecture for Schubert polynomials in Dunkl elements

Let $\widetilde{\cal G}_n^0$ be the quotient of $\widetilde{\cal G}_n$ by the relations $[ij]^2=0$. Let $\widetilde\theta_1,\ldots,\widetilde\theta_n$ be the Dunkl elements, and let $\widetilde{\cal G}_n^+$ be the cone of all nonnegative integer linear combinations of monomials in the generators $x_i$, $1\leq i\leq n$, and $[ij]$, $1\leq i<j\leq n$, of $\widetilde{\cal G}_n^0$. Nonnegativity conjecture. For every $w\in S_n$, the Schubert polynomial $\mathop{\rm S}_w$ evaluated at $\widetilde\theta_1,\ldots,\widetilde\theta_n$ belongs to $\widetilde{\cal G}_n^+$.

This is attributed in the source to [FK], Conjecture 8.1, and concerns positivity of Schubert-polynomial evaluations in the noncommutative Dunkl-element setting. The source gives no resolution.