cosmology - Did really everything begin with a state with very low entropy?

As emphasized by Penrose many years ago, cosmology can only make sense if the world started in a state of exceptionally low entropy. The low entropy starting point is the ultimate reason that the universe has an arrow of time, without which the second law would not make sense. However, there is no universally accepted explanation of how the universe got into such a special state. Are there some observations that would really tell us that the early universe was with small entropy? Is this claim really consistent with our theories?