No-infinite-path conjecture for the uniform infinite meandric system

Let $\mathcal L$ and $\mathcal R$ be independent two-sided simple random walks on $\mathbb Z$ with $\mathcal L_0=\mathcal R_0=0$, and use them to construct the two infinite non-crossing arc diagrams of the uniform infinite meandric system (UIMS). Let the UIMS be the union of these arc diagrams, whose connected components are loops and possibly bi-infinite paths. No-infinite-path conjecture. Almost surely, the UIMS has no infinite path. It is known that almost surely either there is a unique infinite path or there are no infinite paths; determining which alternative occurs is open, and the conjecture asserts the latter.