Markov chain intersections and the loop-erased walk

Let X and Y be independent transient Markov chains on the same state space that have the same transition probabilities. Let L denote the “loop-erased path” obtained from the path of X by erasing cycles when they are created. We prove that if the paths of X and Y have infinitely many intersections a.s., then L and Y also have infinitely many intersections a.s.