Postnikov–Stanley conjecture on the finite Fomin–Stanley algebra

Let $W$ be a Weyl group, let $R^+$ be its positive roots ordered by $\alpha\prec\beta$ when $\beta-\alpha$ is a positive sum of simple roots, and let $J(R^+,\prec)$ be the set of upper order ideals. Let ${\mathbb A}_0$ be the finite nilCoxeter algebra, and let ${\mathbf h}_{i_j}$ denote homogeneous elements of degree $i_j$. Postnikov–Stanley conjecture. The algebra ${\mathbb A}_0$ contains a graded commutative subalgebra ${\mathbb B}'$ satisfying all of the following: over the rationals, ${\mathbb B}'\otimes_{\mathbb Z}{\mathbb Q}$ is generated by homogeneous elements ${\mathbf h}_{i_1},\ldots,{\mathbf h}_{i_r}$ whose degrees are the exponents of $W$; its Hilbert series is

$$P(t)=\sum_{I\in J(R^+,\prec)}t^{|I|};$$

it has a homogeneous basis $\{b_I\mid I\in J(R^+,\prec)\}$ consisting of nonnegative linear combinations of the $A_w$; and the structure constants in this basis are positive. In particular, its dimension is the generalized Catalan number for $W$. This conjecture proposes a uniform finite-Weyl-group analogue of the Fomin–Stanley algebra, with positivity and Catalan enumeration built into its basis and Hilbert series. The source provides no resolution information.