The p-regular product conjecture for coprime conjugacy class sizes

Let $G$ be a group, and let $p$ be a prime. An element of $G$ is p-regular if its order is not divisible by $p$, and the conjugacy class size of an element $x$ is $|G:C_G(x)|$. The p-regular product conjecture. If $x,y\in G$ are two $p$-regular elements with coprime conjugacy class sizes, then $xy$ is a $p$-regular element.

The conjecture would allow results proved for $p$-separable groups to extend to arbitrary groups by removing the $p$-separability assumption in several auxiliary results. The source notes that a partial positive answer had been established elsewhere and that the conjecture has since turned out to be true, although the full result was forthcoming at the time of writing.