Universality conjecture for local topological convergence of network flows

Let $\mathcal{G}^n$ denote the class of embedded graphs in $\mathbb{R}^n$, and let a network flow $G(t)$ have homogeneous initial condition $G(0)\in\mathcal{G}^n$. Let $\sigma_n$ be a probability distribution on countable, connected graphs with a specified root vertex. Universality conjecture for local topological convergence. There exists such a probability distribution $\sigma_n$ such that any such network flow converges in the local topological sense to $\sigma_{\Omega}$ or to a stationary state as $t\to\infty$. The conjecture proposes a universal long-time topological state for homogeneous network flows, while allowing stationary behavior as an alternative; its resolution is not specified in the source.