Meagher–Spiga multipartite derangement graph conjecture

Let $n$ be an even positive integer that is not a power of $2$. A transitive permutation group of degree $n$ has a complete multipartite derangement graph if its derangement graph is complete multipartite.

Multipartite derangement graph conjecture. If $n$ is even but not a power of $2$, then there is a transitive group $G$ of degree $n$ such that its derangement graph is a complete multipartite graph with $n/2$ parts.

This conjecture concerns the existence of transitive groups whose derangement graphs have the specified multipartite structure. The paper constructs infinite families supporting it, including groups of degree $4\ell$ with complete $2\ell$-partite derangement graphs for odd $\ell$, but the assertion for every even $n$ that is not a power of $2$ remains open.