Fomin–Kirillov nonnegativity conjecture for Schubert polynomials

Let $S_n$ be the symmetric group, let $[ij]$ be the generators of the Fomin–Kirillov algebra, and let $\theta_i$ be the Dunkl elements

$$\theta_i=\sum_{j\ne i}[ij].$$

Let $\mathfrak{S}_w$ be the Schubert polynomial associated with $w\in S_n$, and let $\mathfrak{S}_w(\theta)$ denote its evaluation obtained by replacing $x_i$ by $\theta_i$. Nonnegativity conjecture. The element $\mathfrak{S}_w(\theta)$ can be written as a nonnegative linear combination of noncommutative monomials

$$[i_1,j_1][i_2,j_2]\cdots[i_m,j_m].$$

This conjecture, attributed in the source to Fomin and Kirillov, is presented as stronger than the positivity conjecture for ordinary CSM classes and is used to imply that positivity in type A; its resolution status is not given in the source.