Anti-Sidorenko orientations conjecture for trees

Let $T$ be an undirected tree, and call an orientation of $T$ tournament anti-Sidorenko when it has the tournament anti-Sidorenko property.

Anti-Sidorenko orientations conjecture for trees. For every undirected tree, there exists some orientation that has the tournament anti-Sidorenko property. The paper says that this conjecture is affirmatively resolved for a special family of trees, while a complete characterization is not obtained; the general assertion therefore remains open.