Bumpless pipe dreams (BPDs) are combinatorial objects used in the study of Schubert and Grothendieck polynomials. Weigandt recently introduced a co-BPD object associated to each BPD and used them to give an analogue to the change of bases formulas of Lenart and Lascoux between these polynomials. She posed the problem of characterizing the set of permutations whose BPDs have only reduced co-BPDs. We give a pattern-avoidance characterization for these permutations using a set of seven patterns.