One-endedness of essential spanning-forest components in high dimensions

Let $G$ be the infinite nearest-neighbor graph on the integer lattice in $d$ dimensions, and let ${\bf T}$ be the uniform spanning forest obtained as a distributional limit of uniform spanning trees on $d$-dimensional $n$-tori as $n\to\infty$. One-endedness conjecture. If $d\geq 5$, then with probability one each component of the essential spanning forest has only one path to infinity; equivalently, any two infinite paths in a component eventually merge.

This is presented as an open problem because the limiting object in dimensions five and higher is an essential spanning forest rather than a spanning tree of the lattice. The corresponding one-path-to-infinity property is stated earlier for the uniform random spanning tree in dimensions two, three, and four.