This is incorrect, because conventional Benton Rainbow Holograms don't use phase either. That's why they can still operate under white light illumination. The spacing of their interference patterns is unimportant, it only affects the colors of the rainbow artifact. Such an assertion would be silly. This is incorrect.
Mirrors preserve the phase of reflected light.
So do the interference patterns of conventional Rainbow holograms. They preserve the spatially-horizontal phase while scrambling the vertical phase. Their horizontal parallax depends upon their ability to preserve horizontal spatial phase of the illuminating light source. It's like looking into a curved, slit-shaped mirror. We see a virtual image of the illuminating source, and since the illuminator is a point source, we see a glowing dot which floats either behind or before the plastic surface.