Recurrence of nonadaptive random walk in changing environment on subgraphs of the square lattice

Consider a labyrinth evolving by adding edges between adjacent vertices of $\mathbb{Z}^2$, without removing existing edges, and let Theseus perform the random walk on the resulting changing environment. The environment is oblivious if its evolution does not depend on Theseus' location. Recurrence conjecture. If the environment is oblivious of Theseus' location, then Theseus almost surely reaches the exit infinitely many times if he remains in the labyrinth; equivalently, the random walk is recurrent. This conjecture contrasts with the proved possibility of transience when the environment can adapt to the walker's location, and asks whether recurrence is forced in the nonadaptive case.