Recurrence of horizontal–vertical walks

Consider a nearest neighbor random walk on the two-dimensional integer lattice, where each vertex is initially labeled either `H' or `V', uniformly and independently. At discrete time step, walker resamples label at its current location (changing to `V' with probability $q$). Then, it takes mean zero horizontal step if new `H', vertical `V'. This model randomized version of deterministic rotor walk, for which recurrence (i.e., visiting every infinitely often 1) in two dimensions still an open problem. We answer analogous question horizontal-vertical by showing that recurrent $q \in (\frac{1}{3},1]$.