The diameter of strong orientations of Cartesian products of graphs

Let G and H be graphs, and G□H the Cartesian product of G and H. We prove that for every connected bridgeless graphs G and H, the Cartesian product G□H admits an orientation of diameter at most wdiammin(G)+wdiammin(H)+8, where wdiammin(G) denotes the minimum weak diameter of an orientation of G. Orientations of products of graphs that have bridges are considered as well, and an upper bound for the minimum diameter of such orientations is given.