The solubilizer probability bound conjecture

Let $G$ be a finite group and $x\in G$. Define the solubilizer probability of $x$ by

$$P_S(x)=\frac{|\operatorname{Sol}_G(x)|}{|G|},$$

where $\operatorname{Sol}_G(x)$ is the set of elements $y\in G$ such that $\langle x,y\rangle$ is soluble. Solubilizer probability bound conjecture. If $G$ is insoluble, then

$$P_S(x)\leq \frac{3}{5}$$

for every $x\in G$; furthermore, if $x$ is not an involution, then

$$P_S(x)<\frac{1}{2}.$$

Computations for nonabelian simple groups of order at most $2$ million found the value $3/5$ in $\mathrm{A}_5$ and one additional value above $1/2$, namely $5/9$ for an involution in $O(5,3)$. The general bounds remain unproved in the source.