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 $p\ge 5$. 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.