Relate ghost Kohnert poset properties to polynomial properties

Determine whether there are relationships between poset-theoretic properties of \(\mathcal{P}_G(D)\) and algebraic properties of the polynomial \(\mathfrak{G}_D=\sum_{T\in \mathrm{GKD}(D)}\mathrm{wt}^+(T)\).

Background

The paper introduces the naturally associated polynomial $\mathfrak{G}_D$ by summing signed weights over the diagrams in $\mathrm{GKD}(D)$, analogously to the diagrammatic definition of Lascoux polynomials.

The authors leave unresolved whether these polynomials have desirable properties or interpretations and, more specifically, whether their algebraic properties are related to structural properties such as boundedness, lattice structure, rankedness, or join-semilattice structure of $\mathcal{P}_G(D)$.