Classification of orbits under curvature flow for homogeneous topological cloths

Let $\mathcal{G}^n$ be the class of embedded graphs in $\mathbb{R}^n$. Suppose $G\in\mathcal{G}^n$ is an embedded graph with a homogeneous topological cloth and a well-defined time evolution under curvature flow $G(t)$. Classification of orbits conjecture. Either $G(t)$ converges topologically to $\sigma_m$ for some $m\leq n$, or it converges to a stationary state. This conjecture predicts that homogeneous topological cloths have only universal or stationary long-time behavior; the source notes that it would not be surprising if $\sigma_n=\sigma_m$ for all $m,n\geq3$, but gives no resolution status.