The almost simple fixer-ratio conjecture

Let $G$ be an almost simple primitive permutation group, meaning that $G_0\trianglelefteqslant G\leqslant\operatorname{Aut}(G_0)$ for a non-abelian simple group $G_0$, with $G_0=\operatorname{soc}(G)$. Let $\rho_0(G)$ be the fixer-ratio invariant defined by

$$\rho_0(G):=\max\left\{\frac{|K|}{|H|\sqrt{|\Omega|}}:K\leqslant G\text{ is a fixer}\right\}.$$

Almost simple fixer-ratio conjecture. We have

$$\rho_0(G)\leqslant 1$$

for all almost simple primitive groups $G$. This is presented in the source as an equivalent formulation of the bounded fixer-ratio conjecture for primitive groups, via suitable product actions. Its status is open in the supplied source.