Finite forbidden-subdiagram conjecture for shellable Kohnert posets

Let $D$ be a diagram, let $\mathcal{P}(D)$ be its Kohnert poset, and let $\mathcal{F}$ be a finite collection of families of subdiagrams. Finite forbidden-subdiagram conjecture. There exists a finite number of families of subdiagrams $\mathcal{F}$ such that, for every diagram $D$, $\mathcal{P}(D)$ is shellable if and only if there is no $\widetilde{D}\in\mathcal{P}(D)$ containing a subdiagram from $\mathcal{F}$. This conjecture proposes a finite forbidden-subdiagram characterization of shellability, whereas the paper states that a complete general characterization was not obtained.