Dichotomy conjecture for the directed edge-reinforced random walk

Let $G$ be an infinite graph, and let $X_n$ denote the directed edge-reinforced random walk on $G$. For each circuit $C$ in $G$, let $T^C$ be the event that the walk becomes trapped in $C$, and let $d(\cdot,\cdot)$ be the shortest-path distance on $G$.

Dichotomy conjecture. One has

$$\mathbb P\left(\bigcup_{C\in\mathscr{C}}T^C\cup\left\{\lim_{n\to\infty}d(X_n,X_0)=\infty\right\}\right)=1.$$

In other words, the walk either gets trapped in a circuit or escapes to infinity. The conjecture is posed for infinite graphs in light of the preceding discussion and the stated theorem; the supplied text does not indicate whether it has been resolved.