Groups with central 2-sylow intersections of rank at most one

An involution in a finite group is called central if it lies in the center of a 2-Sylow subgroup of G. A 2-Sylow intersection is called central if it is either trivial or contains a central involution. Suppose G is a finite simple group all of whose central 2-Sylow intersections are trivial or rank one 2-groups. It is proved that G is a known simple group.