The primitive derangement subgroup index conjecture

Let $G$ be a primitive permutation group of degree $n$, and let $D(G)$ be the subgroup generated by the derangements in $G$. A Frobenius group is a transitive permutation group whose nonidentity point stabilisers fix no point outside their stabilised point. The primitive derangement index conjecture. If $G$ is not a Frobenius group, then

$$|G:D(G)|\leqslant\sqrt{n}-1;$$

moreover, this bound is attained only if $G$ is an affine group. The bound is proved for non-affine primitive groups, while the remaining case reduces to primitive affine groups.