ROMM is a class of Randomized, Oblivious, Multi-phase, Minimal routing algorithms. Our conjecture is that ROMM routing offers a potential for improved performance compared to fully randomized algorithms under both light and heavy loads. Our conjecture is also that ROMM routing offers close to best case performance for many common permutations. These conjectures are supported by extensive simulations of binary cube networks for a number of routing patterns. We show that k×n buffers per node suffice to make k-phase ROMM routing free from deadlock and livelock on n-dimensional binary cubes.