I attended a talk by Penrose in the ancient world (when travels were a thing, before COVID).

CCC (Conformal Cyclic Cosmology) was definitely an interesting (uncommon) idea. However, CCC does not have the successes of the traditional Big Bang inflationary model, and even have troubles with data. The CCC requires the presence of pre-Big-Bang imprints in large scale structures and in the cosmic microwave background. Those are however not there. I put this link where this is explained in details in an easy-to-grasp manner (there is no way to repeat some points when they are already written very well by someone else).

Cheers, and thanks for this blog!