Mousavi's odd-prime solubilizer conjecture

Let $G$ be a finite insoluble group, let $x\in G$, and write $\operatorname{Sol}_G(x)$ for the set of elements $y\in G$ such that $\langle x,y\rangle$ is soluble. Mousavi's odd-prime solubilizer conjecture. For every odd prime $p$ and every positive integer $n$,

$$|\operatorname{Sol}_G(x)|\neq p^n.$$

The assertion generalizes the corresponding observation for minimal simple groups reported in the paper, but the source supplies no proof for arbitrary finite insoluble groups.