Roger Penrose's conformal cyclic cosmology (CCC) Does the Weyl curvature tensor $C$ of the black hole singularity in the conformal cyclic cosmology diverge to infinity unlike the Big Bang (C = 0)?
 A: Keep in mind that CCC is a dead theory. It became clear by 2010 that it was not viable because it made predictions about particle physics that were not consistent with what we know about particle physics.
The best reference on CCC that I know of, other than the popularization in Penrose's book, is:
R. Penrose, Singularities and time-asymmetry, in General Relativity: An Einstein Centenary Survey, edited by S. W. Hawking and W. Israel, (Cambridge University Press, Cambridge, 1979), p. 581
The way he formulates the hypothesis on p. 630 is:

the Weyl curvature $C_{abcd}$ vanishes at any initial$\ddagger$ singularity.

The footnote says:

That is, with the notation of section 12.3.2, at $\partial \overset{\vee}M$. Note that this includes the final singular TIF of a Hawking black hole explosion -- at which the Weyl curvature indeed vanishes!

So he's not proposing any new physical mechanism that would eliminate black hole singularities in classical GR. Instead, he just points out that it's already believed, if you believe in the methods and results of semiclassical gravity, that quantum effects modify or eliminate the singularities predicted by classical GR.
