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

Let $n$ and $k$ be positive integers, let $\sigma_n$ denote the relevant permutation of the vertex set, and let a $\sigma_n$-$k$-partition be a $\sigma_n$-partition with $k$ parts. 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 $\sigma_n$-$k$-partition with $k>1$ and $n$ odd has a transitive $\sigma_n$-orientation.

This is presented as a stronger conjecture because the authors did not search computationally for partitions with more than three parts; it remains open.