Buratti–Sajna conjecture on resolvable directed cycle decompositions

Let $m$ be a positive odd integer. The complete symmetric digraph $K_{2m}^*$ is the digraph with both directed arcs between every pair of distinct vertices, and a resolvable directed $m$-cycle decomposition is a partition of its arcs into spanning subdigraphs, each a vertex-disjoint union of directed cycles of length $m$.

Buratti–Sajna conjecture. $K_{2m}^*$ admits a resolvable directed $m$-cycle decomposition if and only if $m\geq 5$.

This conjecture completes the determination of the odd cycle-length case of the directed Oberwolfach Problem. The cited work establishes the corresponding existence result for odd $m$ with $5\leq m\leq 49$, while the general case remains open.