Uniform boundedness conjecture for fixers of finite primitive permutation groups

Let $G$ be a finite primitive permutation group on $\Omega$, with point stabiliser $H$, and define

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

Uniform boundedness conjecture. There exists an absolute constant $c$ such that $\rho_0(G)<c$ for all finite primitive permutation groups $G$. This conjecture asserts that the largest fixer, normalized by the order of a point stabiliser and the square root of the permutation degree, is uniformly bounded across all finite primitive permutation groups. Its status is not established in the supplied text.