Favorite sites of a persistent random walk

We consider favorite (i.e., most visited) sites of a symmetric persistent random walk on Z, a discrete-time process typified by the correlation of its directional history. We show that the cardinality of the set of favorite sites is eventually at most three. This is a generalization of a result by Tóth for a simple random walk, used to partially prove a longstanding conjecture by Erdős and Révész. The original conjecture asserting that for the simple random walk on integers the cardinality of the set of favorite sites is eventually at most two was recently disproved by Ding and Shen.