Arc-disjoint hamiltonian paths in non-round decomposable local tournaments

Thomassen proved that a strong tournament T has a pair of arc-disjoint Hamiltonian paths with distinct initial vertices and distinct terminal vertices if and only if T is not an almost transitive tournament of odd order, where an almost transitive tournament is obtained from a transitive tournament with acyclic ordering u1,u2,…,un (i.e., ui→uj for all 1≤i