Homogenization conjecture for weakly reinforced Pólya urns on countable graphs

Let $G$ be a countable graph with uniformly bounded degrees. Let $N_t$ be the system of edge weights initialized at $1$ and evolving according to the continuous-time weakly reinforced Pólya urn dynamics with parameter $0\leq\alpha<1$, and set

$$X_t=\frac{N_t}{t}.$$

An equilibrium is a non-negative vector $\mu\in\mathbb R_+^E$ satisfying

$$\mu(e)=\sum_{v\in e}\frac{\mu(e)^\alpha}{\sum_{e'\in E_v}\mu(e')^\alpha}$$

for every edge $e$ with $\mu(e)>0$. Homogenization conjecture. The equilibrium measure $\mu$ exists, is unique, and $X_t\to\mu$ almost surely. This conjecture asserts that the normalized edge weights of weakly reinforced Pólya urns on every countable uniformly bounded-degree graph converge almost surely to a unique deterministic equilibrium; the supplied source does not indicate whether it has been resolved.