Indeed, mentioned the primordial helium abundance :)
Yup. Well, there's a large degeneracy in the supernova data with regard to the total energy density of the universe. It measures reasonably accurately the ratio of matter (normal + dark) to dark energy, but it doesn't do so well at measuring the total. Because of this, early supernova experiments taken alone were quite consistent with a Milne cosmology (except for the whole issue of the universe not being empty).
But this degeneracy isn't perfect, and more recent supernova experiments which have much more data rule out the Milne cosmology rather well. Of course, once you combine them with something like WMAP or BAO, the Milne cosmology becomes so far out of bounds of observation that the whole idea becomes patently absurd.