Determine whether the existing diagram labeling applies to boundedness

Determine whether the labeling used in Section 4 of \cite{KPuzzle} to compute the maximum number of ghost cells can be applied to establishing when a ghost Kohnert poset \(\mathcal{P}_G(D)\) is bounded.

Background

The authors propose using a labeling of the initial diagram to resolve the unresolved characterization of bounded ghost Kohnert posets. They point to a labeling from Section 4 of \cite{KPuzzle}, which computes the maximum number of ghost cells in a diagram of PG(D)\mathcal{P}_G(D).

It is not established whether that labeling captures the structural information needed to decide boundedness, so its applicability to the boundedness problem is left unresolved.

References

At present, the authors are unaware as to whether this particular labeling can be applied to the problem of establishing when $\mathcal{P}_G(D)$ is bounded.

— Ghost Kohnert posets  (2503.08820 - Hanser et al., 11 Mar 2025) in Final paragraph of Section 3.3, discussion of future work