Solubility conjecture for finite groups with two composite conjugacy class sizes

Let $G$ be a finite group, and let $cs(G)$ denote the set of conjugacy class sizes of $G$. Suppose that exactly two members of $cs(G)$ are composite numbers.

Solubility conjecture. Then $G$ is soluble.

The authors state that solubility is known when $|cs(G)|$ is $3$ or $6$, but they have not proved it when $|cs(G)|$ is $4$ or $5$. The conjecture asserts that the solubility conclusion holds in all cases.