Curvature flow of the octahedron

Let $G$ be the octahedron, and let the simple random walk denote the uniform transition weighting on every vertex. Consider a non-degenerate initial weighting scheme without laziness that is different from the simple random walk.

Octahedral curvature-flow conjecture. The normalized curvature flow on the octahedron converges to a limit described by the stated classification of totally degenerate non-lazy curvature-sharp Markovian weighting schemes.

The proposition preceding the conjecture classifies the relevant limit up to graph automorphism. The convergence assertion is based on the structure identified there and is not established in the supplied text.