Fomin–Kirillov nonnegative Schubert-polynomial conjecture in the Fomin–Kirillov algebra

Let $\mathcal{E}_n$ be the Fomin–Kirillov algebra generated by $\mathbf{x}_{ab}$ for $1\leq a<b\leq n$, and define the commuting Dunkl elements

$$\theta_i=-\sum_{a<i}\mathbf{x}_{ai}+\sum_{i<b}\mathbf{x}_{ib}.$$

Let $\mathfrak{S}_w$ denote the Schubert polynomial of $w\in\mathfrak{S}_n$, and let $\mathcal{E}_n^+$ be the cone of nonnegative integer linear combinations of noncommutative monomials in the generators. Fomin–Kirillov conjecture. For every $w\in\mathfrak{S}_n$,

$$\mathfrak{S}_w(\theta_1,\ldots,\theta_n)\in\mathcal{E}_n^+.$$

The source presents this as implying the non-equivariant CSM positivity conjecture in type $A$; it does not report a general resolution.