Separation of two pointed polyhedral cones using hyperplanes generated by facets

Question

Let $C_1$ and $C_2$ two pointed (that is, with vertex in $0$) polyhedral cones in $\mathbb{R}^n$ with $\dim(C_1)=\dim(C_2)=n$. If $$\mbox{relative interior}(C_1)\cap \mbox{relative interior}(C_2)=\emptyset,$$ then it is clear that there exists a hyperplane that separates $C_1$ and $C_2$. My question is: Does there exist a $n-1$-dimensional face $F$ of $C_1$ or $C_2$ such that $F$ generates a hyperplane $H$ that separates $C_1$ and $C_2$? That is, is it possible to take as hyperplane of separation one of the hyperplanes generated by the faces of dimension $n-1$ of $C_1$ or $C_2$?

The answer is yes if $C_1\cap C_2$ has dimension $n−1$, since $C_1\cap C_2$ is then included in some face of dimension $n−1$ of $C_1$ or $C_2$. But if $C_1\cap C_2$ has dimension $<n−1$ then it is not clear.

Answer 1

Yes for dimension 2 (pick the cone with larger angle; one of its sides fits). No for larger dimensions.

For a counterexample in dimension $3$, let $C_1$ be a positive orthant, and let $C_2$ be a negative orthant rotated by $\pi/3$ around $x=y=z$. (Note that you need to check just the faces of $C_1$, since the cones can be swapped by an orthogonal transform.)

There are similar examples in any higher dimension.