Find a general necessary condition for bounded Kohnert posets

Determine a general necessary condition for a Kohnert poset to be bounded.

Background

The paper contrasts ghost Kohnert posets with ordinary Kohnert posets. For ghost Kohnert posets, it proves a necessary condition for boundedness in terms of the initial seed diagram.

In the cited literature, a sufficient condition for boundedness of a Kohnert poset is known, but the authors state that no general necessary condition was known at the time of writing. This is an explicitly identified unresolved problem concerning ordinary Kohnert posets.