Maximizing return time for a 2-d random walk

Question

Consider a 2-d grid with m rows and n columns. The decision variables are transition rates, subjecting to the following constraint: (jump rate to the right adjacent node)+(jump rate to the above adjacent node)=1; (jump rate to the left adjacent node)+(jump rate to the below adjacent node)=1; the left-most up-most corner node is positive recurrent and there is only one communication class. The goal is to maximize the hitting time, starting from the left-most up-most corner node, until hitting itself again. Is there a clean answer to this question?