Dark energy is not an illusion after all: They accept that the expansion rate in the local universe is higher than in more distant regions. But instead of assuming the expansion rate has increased with time, they suggest our patch of the universe happens to contain less matter than average. Within this "void", the expansion rate is higher than outside because there is less gravity to slow it down.
But new, more precise measurements of supernovae, taken by the Hubble Space Telescope, clash with the simplest version of the void model. That model could be made to fit previous supernova measurements and other cosmological data, but only if the local expansion rate is about 60 kilometres per second per megaparsec or less.