Not from an external point of view, as you might have a momentum component perpendicular to that plane

(but yes I think you might be right if we're centered on the CG)

One of the necessary conditions here is that the three objects return to their exact initial position, and so does the centre of mass. Initial conditions with non-zero momentum must trivially be ruled out. But this doesn't stop them from having velocities perpendicular to the initial plane that cancel out perfectly, so this doesn't refute the assertion that the planes keep changing.