Path formation conjecture for vertex-reinforced non-backtracking random walks

Let $X$ be a vertex-reinforced non-backtracking random walk on a connected finite undirected graph, with reinforcement function $W(k)=(1+k)^\alpha$. Let $C=\{i_1,\dots,i_\ell\}$ be a random path as in Proposition 0.1, and let $L$ denote its length. Path formation conjecture. If $\alpha>3$, then almost surely there exist such a path $C$ and $k_0\geq 1$ such that, for every $k\geq k_0$ and every $\ell\in\{1,\dots,L\}$,

$$X_{kL+\ell}=i_\ell.$$

The conjecture predicts eventual periodic traversal of a single random path; the preceding results establish this behavior in the complete-graph setting, while the general connected-graph case remains open.