I have a convex cone $H$ that is the convex hull of two other convex cones $F$ and $G$. Furthermore, for all $f\in \mathsf{faces}(F)$, $g\in \mathsf{faces}(G)$ we have that $\mathsf{conv}(f,g)\in \mathsf{faces}(H)$ (e.g. the convex hull of any face of $F$ with any face of $G$ is also a face). Does this imply that $H$ is decomposable into $H=F\oplus G$ (e.g. $G$ and $F$ have disjoint spans)?
It seems that it should imply this by some simple dimensional argument..