Separation of two cones using hyperplane generate by some facet.

Question

Let $C_1$ and $C_2$ two polyhedral cones (pointed, that is, with an only vertex on $0\in R^n$), we suppose that $\dim(C_1)=\dim(C_2)=n$. If $$\mbox{relative interior}(C_1)\cap \mbox{relative interior}(C_2)=\emptyset,$$ is clear that there exist a hyperplane that separate $C_1$ with respect to $C_2$. My question is, exist a $(n-1)$-dimensional face $F$ of $C_1$ or $C_2$ such that $F$ generate a hyperplane $H$ that separate $C_1$ with respect to $C_2$?. That is, is possible that I can take as hyperplane of separation some of the hyperplanes generate by some of the face of dimension $(n-1)$ on $C_1$ or $C_2$.