Mostly its controversial simply because the evidence for it is week. You need to perform a complicated analysis of the CMB, to find some statistical artifacts that are consistent with your theory, and not any other theory. This is incredibly difficult, because the CMB is very noisy, we don't have a full understanding of how the universe evolved post big bang, and even with our current understanding, a full simulation is computationally impossible. Penrose's analysis found the signature that shows CCC, however when others attempt to perform the analysis, they do not find it. As such, the accuracy of the only evidence remains controversial.

In addition to the empirical lack of convincing evidence, there is also a theoretical problem. In order for CCC to work, you need to assume that all massive particles decay. The is not inconceivable, but involves things like proton decay, which is speculative at best, and electron decay, for which (as far as I am aware) we do not even have a serious speculative mechanism.