Perfect-matching minimum-action conjecture for complete graphs

Let $K_n$ be a complete graph with an even number of vertices, and let the action be the sum of the edge curvatures. A perfect-matching minimum-action conjecture states that the minimum action for $K_n$ is achieved by the perfect-matching setting, in which the action is $n$, the number of vertices. This is proposed in analogy with the tree case, while the preceding theorem establishes the maximum action for the constant edge-length setting.