Characterize bounded ghost Kohnert posets

Characterize, entirely in terms of an initial ghost-free diagram D, exactly when the ghost Kohnert poset \(\mathcal{P}_G(D)\) is bounded, thereby characterizing when it is a lattice.

Background

The paper proves that every ghost Kohnert poset $\mathcal{P}_G(D)$ associated with a diagram having no ghost cells is a ranked join-semilattice. Consequently, such a poset is a lattice exactly when it is bounded.

The authors establish only a necessary condition for boundedness: the free cell sequence of the initial diagram must be strictly increasing. They then give an example showing that this condition is not sufficient, so a complete characterization remains unresolved. The authors suggest that a suitable labeling of the diagram might help produce the missing characterization.