One-prime power hypothesis for conjugacy class sizes

A finite group G satisfies the one-prime power hypothesis for conjugacy class sizes if any two conjugacy class sizes m and n are either equal or have common divisor a prime power. Taeri conjectured that an insoluble group satisfying this condition is isomorphic to S × A where A is abelian and S ∼= P SL₂(q) for q ∈ {4, 8}. We confirm this conjecture.