Orbit sizes for complements of paths

Prove that if P_m is a path graph with m vertices, G is its n-vertex complement, m is odd, and n is even, then every orbit of the refractive toric promotion map on G has size either 6 or 6(mn-n-3m+4).

Background

The paper introduces refractive toric promotion as the finite-state dynamical system encoding the billiard trajectories and studies ensnaring behavior rather than orbit cardinalities. Computational and structural observations led the authors to identify complements of odd paths with even total vertex count as a family having apparently constrained orbit sizes. The stated conjecture predicts the only two possible sizes.