Yes, though one has to go through quite a bit of trouble to actually show that. The big problem is, that speed of light is defined with reference to a backgroun

Yes, though one has to go through quite a bit of trouble to actually show that. The big problem is, that speed of light is defined with reference to a background geometry and gravitational waves are a distortion of this background geometry. (And this has of course not be shown yet, however this observation, if the rumors are true, would be evidence for that.)