Intersections of blocks of cyclotomic Hecke algebras

Trinh and Xue have proposed a startling conjecture on intersections of blocks of cyclotomic Hecke algebras occurring in modular representation theory of finite reductive groups. We prove this conjecture for all exceptional type groups apart from a few situations in type E₈ E 8 . We also give a conceptual proof in all cases where relative Weyl groups are cyclic. Furthermore, we propose several generalisations, to Suzuki and Ree groups, to non-rational Coxeter groups and even more generally to spetsial complex reflection groups, and confirm these in various cases.