This space is not curved, but the time dilation/index of refraction can be determined by observing the difference in clock rates at different gravitational potentials. It is also not empty, as it contains the CMBR and the gravitational field.
As for speculation on how I think space will be modeled in the final form of this theory, eventually I hope to map pseudospace-time to the Hurwitz integers (on the Plank scale), paired with a space-pseudotime phase space over the algebra of the split-octonions. I have not yet figured out how to make this scale shrink. This is because I also expect the redshift to eventually be explained as an increase in the fine structure constant with the Hubble expansion. This paragraph is speculation, in response to a direct question, and not part of the theory being defended here.
Edit: The linked article suggests that the final lattice must be much smaller than I suggested above.