Morris–Spiga–Roney-Dougal conjecture on intersection densities of transitive groups

Let $G$ be a transitive permutation group of degree $n$. For a transitive permutation group $K$ of degree $n$, let $I(n)$ be the maximum intersection density among such groups. If $n$ is even but not a power of $2$, a transitive group $H$ of degree $n$ should exist whose derangement graph $\Gamma_H$ is a complete multipartite graph with $n/2$ parts. If $n$ is a prime power, then $I(n)=1$. If $n=pq$ for odd primes $p>q$, then $I(n)=1$. If $n=2p$ for a prime $p$, then $I(n)=2$. Morris–Spiga–Roney-Dougal conjecture. These four assertions describe the expected intersection densities and derangement-graph structure for transitive permutation groups in the stated degree classes. Parts (ii) and (iv) have been settled, respectively, in work of Hilton, Koo, Meagher, and Mütze and of Roney-Dougal; the remaining assertions are not resolved here.