Finite forbidden-subdiagram characterization of ranked Kohnert posets

A diagram $D_0$ determines a Kohnert poset $P(D_0)$, and $KD(D_0)$ denotes the collection of diagrams associated with $D_0$. A diagram $D$ contains a subdiagram from a family of subdiagrams $F$ when at least one member of $F$ occurs as a subdiagram of $D$.

Finite forbidden-subdiagram conjecture. There exists a finite number of special families of subdiagrams $\mathcal{F}$ such that, for any diagram $D_0$, $\mathcal{P}(D_0)$ is ranked if and only if there is no $D\in KD(D_0)$ such that $D$ contains a subdiagram from $\mathcal{F}$.

This conjecture proposes a finite forbidden-subdiagram characterization of rankedness for Kohnert posets. The article establishes necessary conditions and rankedness results for several special families, but a general sufficient characterization remains open.