Monotonicity of Recurrence in Random Walks

We consider non-homogeneous random walks on the positive quadrant in two dimensions. In the 1960's the following question was asked: is it true if such a random walk X is recurrent and Y is another random walk that at every point is more likely to go down and more likely to go left than Y, then Y is also recurrent? We provide an example showing that the answer is negative. We also show that if either the random walk X or Y is sufficiently homogeneous then the answer is in fact positive.