Solubility from a disconnected common divisor graph on p-regular conjugacy class sizes

Let G be a finite group and let p be a prime. We prove that if the common divisor graph on the nontrivial conjugacy class sizes of p-regular elements of G is disconnected, then G is soluble. This answers positively the question left open by Camina, Maróti, Pacifici, Parker, Rekvényi, Saunders, Sotomayor, Tracey and van Beek: whether disconnectedness forces solubility without a p-solubility hypothesis.