  Cosmic censorship protects the outside world from black hole singularities
and paves the way for assigning entropy to gravity at the event horizons. We
point out a tension between cosmic censorship and the quantum backreacted
geometry of Schwarzschild black holes, induced by vacuum polarization and
driven by the conformal anomaly. A similar tension appears for the Weyl
curvature hypothesis at the Big Bang singularity. We argue that the requirement
of exact conformal symmetry resolves both conflicts and has major implications
for constraining the set of fundamental constituents of the Standard Model.
