Maximum-action conjecture for triangle and square lattice graphs

Let a triangle or square lattice graph be equipped with edge lengths, and consider the action obtained by summing the Ollivier-Ricci curvatures over its edges. The maximum-action conjecture states that the constant edge-length setting gives the maximum action. The paper notes that direct computation gives action zero for the constant edge-length setting; whether this setting is globally maximal is left as a conjecture.