I find plane waves are uncompatible with light cone.
Perhaps plane waves are "virtual" and can never be measured in that case, shouldn't we call plane waves as "virtual plane waves"? (other option could be that plane waves allows waves travel faster than c)
I would like to clarify this point through this question.
If plane waves would exist(as measurable), then higher than c speed could be reached like this:
A wave going from X to Y at a speed c, it will reach Z at higher than c speed, because it will reach at same time, traveling more distance.
(X). | v ________________________________________________ plane waves ________________________________________________ going X to Y ________________________________________________ (Y). (Z).
In a real situation the wave will be a circle (or a sphere in 3d) so it will get Z later then that's not a plane wave.