The odd-order obstruction conjecture for transitive n-partition orientations

Let $n$ be a positive integer, let $\sigma_n$ denote the relevant permutation of the vertex set, and let a $\sigma_n$-partition be a partition whose parts are compatible with $\sigma_n$. A transitive $\sigma_n$-orientation is a transitive orientation of the complete graph that is a $\sigma_n$-orientation of the partition.

Odd-order obstruction conjecture. No $\sigma_n$-partition with $n$ odd has a transitive $\sigma_n$-orientation.

The authors found transitive $\sigma_n$-orientations only for even $n$ and report that computer searches failed to find examples for odd $n$; the conjecture remains open.