The book of course deals with CCC (Conformal Cyclic Cosmology), developed by Penrose. The key idea is that a universe heavily expanded under dark energy turns out to be so homogeneous as the very first moments after the hot and dense Bing Bang. This equivalence is named conformal, meaning that the shapes are preserved. I can more or less see the idea but I don't understand how the universe would go from the mathematical duality to the physical realisation.
Anyway, the idea needs 2 things that Afshordi says the community is finding difficult to accept. First, in the hyper faraway future all the matter should have been converted into radiation. This is problematic because some particles are considered forever stable, like the electron. But perhaps the last word on particle physics hasn't been uttered.
But the second necessity is even worse. CCC needs to break the unitarity of quantum mechanics, so the information is lost from one cycle to the other. I wonder if all cyclical cosmologies need this. This means that General Relativity > Quantum Mechanics. This is a minority position, although it's not that Penrose is alone either.