Fomin–Kirillov additive-generation conjecture for Dunkl evaluations

Let $S_n$ be the symmetric group, let $\mathcal{E}_n^+$ be the positive cone generated by noncommutative monomials in the Bruhat generators $[ij]$, and let $\theta_1,\dots,\theta_n$ be the Dunkl elements in $\mathcal{E}_n$. Let $C$ be the commutative subalgebra generated by the Dunkl elements. Fomin–Kirillov additive-generation conjecture. The evaluations $\mathfrak{S}_w(\theta)$, for $w\in S_n$, are the additive generators of

$$\mathcal{E}_n^+\cap C.$$

The claim proposes an alternative description of the basis of Schubert cycles. It refines the positivity framework by characterizing all positive elements lying in the commutative Dunkl subalgebra.