Non-constant-solution nonexistence conjecture for the tEoM on $T_2$

Let $T_2$ be the graph appearing in the paper, and let a setting be an assignment of edge lengths satisfying the tEoM. The non-constant-solution nonexistence conjecture states that there exists no non-constant solution to the tEoM on $T_2$ when all edge lengths are positive. This is the second conjecture in the paper's discussion of positive solutions and remains unproved in the supplied text.