Equivariant Coxeter-complex surjection conjecture for the superspace coinvariant ring

Let $W$ be a finite reflection group of rank $n$, let $R^{(1\mid1)}_W$ be its superspace coinvariant-type quotient, and let $\mathcal{OP}_W$ be the Coxeter complex formed by the faces cut out on a sphere by the reflecting hyperplanes. Write $\mathcal{OP}_{W,k}$ for its $k$-dimensional faces, let $\mathbb{R}[\mathcal{OP}_W]$ and $\mathbb{R}[\mathcal{OP}_{W,k}]$ be the corresponding permutation modules, and let $\det$ be the one-dimensional determinant representation of $W$. Equivariant Coxeter-complex surjection conjecture. There exists a $W$-equivariant surjection

$$R^{(1\mid1)}_W\twoheadrightarrow\mathbb{R}[\mathcal{OP}_W]\otimes\det,$$

and, for every $k$, there is a $W$-equivariant surjection

$$\bigl(R^{(1\mid1)}_W\bigr)_{*,n-k}\twoheadrightarrow\mathbb{R}[\mathcal{OP}_{W,k}]\otimes\det.$$

The conjecture proposes an equivariant refinement of the observed relationship between the fermionic Hilbert series and the reversed $f$-vector of the Coxeter complex; the text notes agreement in several types but a discrepancy in type $F_4$, and leaves the asserted surjections open.