If an object actually occupies less space physically when moving, it cannot depend on the direction we define as positive. As we know, an object aligned with the direction of relative motion is contracted whether it is fixed in frame S and viewed from S', or the other way around. Use this idea to argue that distances along the y- and y'-axes cannot differ at all. Consider a post of length Lo fixed in frame S, jutting up from the origin along the +y-axis, with a saw at the top poised to slice off anything extending any higher in the passing frame S'. Also consider an identical post fixed in frame S'. What happens when the origins cross?
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