Neumann's fixed-point conjecture for primitive permutation groups

Let $G$ be a finite primitive non-regular permutation group of degree $n$. Write $\operatorname{fix}(g)$ for the number of points fixed by $g\in G$. Neumann's conjecture. There exists $g\in G$ such that

$$1\leq \operatorname{fix}(g)\leq n^{1/2}.$$

The conjecture strengthens Neumann's result for non-regular transitive groups, which guarantees an element fixing between $1$ and $n/2$ points. The source notes that most work has focused on primitive groups of affine type.