I do not understand most of that paper, and probably never will, but can someone explain why they propose cubes?

If I take a crate of tennis balls, and compress it, I expect the balls to first (more or less) form a densest sphere packing, and from there, I expect they will start to deform to look like truncated octahedrons (http://en.wikipedia.org/wiki/Bitruncated_cubic_honeycomb)

Those may be cubic, but it aren't cubes.

Also, since there is no unique densest sphere packing, it would not surprise me if variations on this are possible (http://en.wikipedia.org/wiki/Sphere_packing).