Let $G$ be a finitely generated hyperbolic group with the word metric; fix a symmetric generating set $S$ and let $\mathcal{G}$ be the Cayley graph of $G$ w.r.t. $S$. Define the cone of an element $x\in G$ as the set of all $z\in G$ such that there exists a geodesic in $\mathcal{G}$ starting from (the vertex correspondent to) $e$, passing through $x$ and terminating in $z$. Denote the cone type of $x$ with $C(x)$.

Pick two elements $x_1,x_2\in G$. My question: can we write the intersection $C(x_1)\cap C(x_2)$ as a finite union of cones?