Reduced TAS-tree conjecture for trees without isomorphic pairs

An undirected tree is a connected graph with no cycles. A caterpillar is a tree for which removing all leaves results in a path. An isomorphic pair in a graph consists of disjoint vertex sets inducing isomorphic subgraphs and satisfying the specified two-edge cut condition. An orientation has the tournament anti-Sidorenko property (TAS) when it satisfies the tournament anti-Sidorenko inequality.

Reduced TAS-tree conjecture. For every undirected tree that is not a caterpillar, has at least $2$ vertices of even degree, and has no isomorphic pair, there exists some orientation that has the tournament anti-Sidorenko property.

Caterpillars and trees reducible by isomorphic-pair pruning are already covered by results in the paper, so this conjecture is the remaining reduced case of the TAS-tree problem.