The hsop existence conjecture for well-generated reflection groups

Let $W$ be an irreducible well-generated finite reflection group, with Coxeter number $h=d_n$, and let $S$ be its polynomial ring, $V$ its reflection representation, and $V^*$ its dual. An hsop is a homogeneous system of parameters in $S$.

The hsop existence conjecture. If $p\equiv 1\pmod h$, then $S$ contains an hsop $\mathbf{\Theta}$ carrying $V^*(-p)$; if $p\equiv -1\pmod h$, then $S$ contains an hsop $\mathbf{\Theta}$ carrying $V(-p)$.

This conjecture supplies the hsops needed for the preceding Hilbert-series formulas and connects invariant-theoretic constructions with the $W$-Catalan and cyclic-sieving phenomena. The source gives no resolution status.