I can't say I understood all of it, but this I got.
In it [De revolutionibus orbium coelestium], Copernicus (among other things) put forth the concept that the Earth is not unique and does not occupy any sort of special position within the solar system. A generalization of this principle—the earth does not occupy a favored position within the universe—has, along with Einstein's equations, become the founding assumption of modern cosmology.I think it's time for me to bury my head in the sand and deny any of this is happening.
In the upcoming PRL paper, the authors postulate that we are indeed in a special location within the universe, specifically, "near the centre of a void where the local matter density is low." This isn't exactly a small void—it would need to be on the order of the size of the visible universe to get the new model to work. Still, the earth would be near its center, which is a vaguely pre-Copernican notion and, as a side effect, the model does away with dark energy.