Proving a conjecture for fusion systems on a class of groups

We prove the conjecture that exotic and block-exotic fusion systems coincide holds for all fusion systems on exceptional p- groups of maximal nilpotency class, where p >= 5. This is done by considering a family of exotic fusion systems discovered by Parker and Stroth. Together with a previous result by the author, which we also generalise in this paper, and a result by Grazian and Parker this implies the conjecture for fusion systems on such groups. Considering small primes, there are no exotic fusion systems on 2-groups of maximal class and for p = 3, we prove block-exoticity of two exotic fusion systems described by Diaz-Ruiz-Viruel.(c) 2023 Elsevier Inc. All rights reserved.