From what I know about LQG, it seems more likely that the PBS spacetime interpretation is probably the right one.
In standard general relativity, spacetime is considered to be smooth and continuous. That is, you can in theory divide it into smaller and smaller volumes without limit. Loop quantum gravity, LQG on the other hand (because it is a quantum theory after all), breaks this smoothness into a discrete structure where spacetime is a spin network of lines and nodes.
Loop quantum gravity predicts that the speed of light is not constant which violates one of the fundamental premisesconcepts of relativity. Specifically, higher energy photons shouldwould travel slower than lower energy photons.
Low frequency light, for example radio waves, have long wavelengths. For such a wavelength, travelling through spacetime will be fairly "smooth" since there is no microscopic structure of spacetime to contend with, since the scale for a radio wave is so much bigger than any loop quantum gravity scale. But on the opposite side, for example a gamma ray photon, has a very high frequency and a very small wavelength, one which exists on a scale somewhat closer to the scale where LQG (or any quantum theory of gravity) effects will become more noticeable.
So one could conclude that light of high energy and frequency, as it travels through spacetime interacts with these microscopic points in the spin network, so that the net effect is a slowed down light ray, and as stated above, the converse is true for low energy long wavelength light, who's interaction with the same is negligible.
I'm inclined to think, that although there is this description of changing light-speed at different scales, one could still say that the local speed of light still remains unchanged. But there are various physicists that actually think that if LQG were accurate, and such spin networks existed, this would break the notion of the constant nature of the speed of light.