One-prime power hypothesis for conjugacy class sizes

A finite group G satisfies the on-prime power hypothesis for conjugacy class sizes if any two conjugacy class sizes m and n are either equal or have a common divisor a prime power. Taeri conjectured that an insoluble group satisfying this condition is isomorphic to StimesA where A is abelian and ScongPSL2(q) for qin4,8. We confirm this conjecture.