On intersections of Sylow 2-subgroups in automorphism groups of finite simple groups of exceptional Lie type

It is shown that, in a finite group with socle isomorphic to a simple group of exceptional Lie type over a field of an odd order greater than 3, the number of Sylow 2-subgroups whose intersection with a fixed Sylow 2-subgroup is trivial exceeds by a factor of at least 3 the order of this Sylow 2-subgroup.