A couple things on conformal cyclic cosmology (CCC) for smooth 3+1 dimensional spacetimes:
> it restarts once there's nothing left but energy
The "nothing left but energy" part is a requirement that matter (in the broadest sense) must be conformally invariant so that its active gravitation as a source allows for the CCC conformal rescaling of the FLRW metric, which is how a predecessor "aeon" is connected mathematically to its successor.
Of the Friedmann dusts, only lightlike radiation can be rescaled this way. Equivalently, all the matter left in the universe at the end of an aeon must be on null geodesics.
Because local Lorentz invariance is fully baked into the Standard Model of Particle Physics (SM), and because the SM has particles with nonzero rest masses (which in a Lorentzian patch cannot couple to null geodesics), NOBODY knows how to do this in a consistent way (let alone in a way that matches actual evidence from particle physics).
(If we restrict to QED we need to convert all electrons and positrons into photons and impose some unknown mechanism to suppress statmech fluctuations back to a non-conformally-invariant condition. In reverse order, two-photon physics are maaaaaaybe supressible adiabatically. However, there just aren't enough positrons to find and annihilate every electron (adiabatic expansion makes that even less likely!), and electrons on their own don't decay into photons. Of course, once we add in the weak and strong forces, we are far beyond quantum electrodynamics, with all sorts of new ways in which reaching a conformally invariant stress-energy state becomes implausible. Oh yeah, and now do dark matter.)
WRT previous comments in this thread, any black hole (BH) at the end of an "aeon" must radiate only massless bosons. This puts a pretty strong lower limit on the mass of Hawking-radiating BHs crossing an boundary between "aeons", or destroys the mathematics of the hypothesized conformal rescaling by virtue of having incompatible non-conformally-invariant field theories on both sides of the Einstein Field Equations.
Binary BHs don't fit cleanly into the FLRW rescaling picture either: among other things they source a metric that isn't locally isotropic and homogeneous (a dust of high mass isolated singleton Birkhoff-theorem BHs is mostly fine though, and in principle you could get through through BH mergers and a stronger cosmological coupling (some types of "fifth force"/quintessence, for instance, to break apart wide BH binaries/triples/multiples)).
There are maybe escapes from some of these constraints in extra dimensions and lattices, but I've been under the impression that one of the attractions of CCC is that it's compatible with the FLRW metric of the standard cosmology.
Does CCC work with small perturbations on the boundary between aeons? Who knows. However, it probably works with small perturbations near that boundary, because we do perturbative FLRW routinely these days. So maybe there's some mechanism (e.g. in dark energy) that makes small deviations from conformally invariant field configurations entirely vanish at the aeon boundary. But we have no astrophysical or particle physics evidence for that at all.
Finally, the article at the top is about inhomogeneous cosomology -- i.e., Timescapes metric is not FLRW metric -- and whatever one's position on CCC, it rests on the standard view that at large scales the universe is homogeneous and that any inhomogenities, anisotropies, and backreactions vanish at smaller scales (and so admit perturbation theory).