The unique-obstruction conjecture for Hamilton decompositions of regular tripartite tournaments

Let $n$ be sufficiently large, and let $G$ be a regular tripartite tournament on $3n$ vertices. Denote by ${\mathcal T}_\triangle$ the known regular tripartite tournament obtained from $\overrightarrow{C_3}(n)$ by reversing the edges of a single triangle. Unique-obstruction conjecture. Every regular tripartite tournament on $3n$ vertices, except for ${\mathcal T}_\triangle$, has a Hamilton decomposition. This conjecture identifies the sole known obstruction to Hamilton decompositions in regular tripartite tournaments; the paper's main theorem proves an almost-decomposition result, while the conjecture itself remains open.