Hamaker–Pechenik–Weigandt's CDG Gröbner basis pattern-avoidance conjecture

Let $w\in S_n$ be a permutation. The CDG generators are the diagonal Gröbner generators introduced for the matrix Schubert ideal $I_w$, and a permutation avoids a pattern if it contains no occurrence of that pattern.

Hamaker–Pechenik–Weigandt's conjecture. The CDG generators are a diagonal Gröbner basis for $I_w$ if and only if $w$ avoids all eight patterns

$$13254,\ 21543,\ 214635,\ 215364,\ 215634,\ 241635,\ 315264,\ 4261735.$$

This conjecture gives a pattern-avoidance criterion for when the proposed diagonal Gröbner generators work. The present paper states that it proves the conjecture, so its status is solved.