The uniform insolubility conjecture for finite non-abelian simple groups

Let $N$ be a finite non-abelian characteristically simple group, let $a,b\in\operatorname{Aut}(N)$, and identify $N$ with $\operatorname{Inn}(N)$. Define $\operatorname{P_{ins}}(N,a,b)$ as the probability that a uniformly chosen $n\in N$ satisfies that $\langle a,bn\rangle$ is insoluble. Say that $N$ is $\eta$-insoluble, for a positive real number $\eta$, if

$$\operatorname{P_{ins}}(N,a,b)>\eta$$

for every $(a,b)\in\operatorname{Aut}(N)$ with $a\ne 1$. Uniform insolubility conjecture. There exists a positive real number $\eta$ such that every finite non-abelian simple group is $\eta$-insoluble. The paper's main theorem proves the solubilizer characterization for profinite groups whose non-abelian composition factors are uniformly $\eta$-insoluble. This conjecture would remove that hypothesis and supply the uniform finite-simple-group input needed for the general case.