Invariant-measure universality conjecture for rescaled curvature flow

Let $\hat{\Omega}$ be curvature flow on embedded graphs in $\mathbb{R}^n$, rescaled so that a characteristic length scale remains equal to one, and let $\mathcal{G}^n$ be the class of embedded graphs in $\mathbb{R}^n$. Invariant-measure universality conjecture. There exists a unique $\hat{\Omega}$-invariant, homogeneous probability measure $\nu$ on $\mathcal{G}^n$ such that $\nu$-almost all graphs do not evolve to a stationary state under $\hat{\Omega}$; furthermore, $\hat{\Omega}$ is ergodic with respect to $\nu$. Such a measure can be obtained from any homogeneous initial condition $G\in\mathcal{G}^n$ that does not converge to a stationary state by taking the limit of $\mathcal{P}(G(t))$ as $t\to\infty$. This reformulates the proposed universality behavior in terms of a unique invariant ergodic measure, but the source does not specify a resolution status.