Pattern characterization conjecture for shellable Kohnert posets of key diagrams

Let $\mathbf{a}=(a_1,\ldots,a_n)$ be a weak composition, and let $\mathcal{P}(\mathbb{D}(\mathbf{a}))$ be the Kohnert poset of its key diagram. Pattern characterization conjecture. The poset $\mathcal{P}(\mathbb{D}(\mathbf{a}))$ is shellable if and only if there are no indices $1\leq i_1<i_2<i_3\leq n$ satisfying either $a_{i_1}<a_{i_2}<a_{i_3}$ or $a_{i_1}\leq a_{i_3}-3\leq a_{i_2}-3$, and there are no indices $1\leq j_1<j_2<j_3<j_4\leq n$ satisfying any of $a_{j_1}\leq a_{j_2}<a_{j_3}-1\leq a_{j_4}-1$, $a_{j_1}\leq a_{j_2}<a_{j_4}<a_{j_3}$, $a_{j_2}<a_{j_1}<a_{j_4}<a_{j_3}$, or $a_{j_2}<a_{j_1}<a_{j_3}\leq a_{j_4}$. This conjecture seeks a complete pattern-avoidance characterization without the purity restriction; the paper proves the necessity of these avoidance conditions and conjectures sufficiency.