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I don't think I would use UM-style linear + torsion rods for this. That's definitely a lot of angular momentum. Probably go with a lightweight cable sync system like synchromesh, or smaller (5mm?) torsion rods like you see in Replicator 1 style Y stages. The benefit to synchromesh is that the sync and drive cables for each direction could be one continuous run, if you don't mind a lot of idler pulleys.
You could put the four Z stages on screws instead of belts, if you want.