Constant-solution conjecture for positive-edge-length tEoM settings on $T_q$

Let $T_q$ be the graph under consideration, let $M_q$ be a setting of its edge lengths, and let the tEoM denote the discrete time equations of motion. Assume that all edge lengths in $M_q$ are positive and that there is a vertex $i\in T_q$ at which all connected edges have equal lengths. The constant-solution conjecture states that if $M_q$ solves the tEoM, then $M_q$ is a constant solution. The conjecture concerns existence and persistence of positive solutions; the paper motivates it by observing that evolution away from an equal-edge vertex should otherwise produce nonpositive edge lengths after finitely many steps.