  We study entire continuous viscosity solutions to fully nonlinear elliptic
equations involving the conformal Hessian. We prove the strong comparison
principle and Hopf Lemma for (non-uniformly) elliptic equations when one of the
competitors is $C^{1,1}$. We obtain as a consequence a Liouville theorem for
entire solutions which are approximable by $C^{1,1}$ solutions on larger and
larger compact domains, and, in particular, for entire $C^{1,1}_{\rm loc}$
solutions: they are either constants or standard bubbles.
