Version 3 (June 10, 2026): the manuscript prepared for journal submission, after an adversarial verification pass that confirmed every lemma, forcing triple, and case of the proof. Changes vs v2: the abstract's description of the exceptional cases is corrected (three pairs, (4,1), (4,2), (5,1)); an explicit color-swap reduction covers both configurations in the cascade proofs; three small arithmetic justifications in the cascade are repaired; a new remark shows the hypothesis b >= 4 is necessary (with a valid b=2 coloring); the refinements (sharp failure set and moduli of valid colorings) are moved after the structure lemma's proof; and citations are corrected (Gupta-Thulasi Rangan-Tripathi, Ann. Comb. 19(2):269-291; Boza-Marin-Revuelta-Sanz).
Alexander Towell (Wed,) studied this question.