Let $G$ be a locally compact Hausdorff topological group.
Definition A closed normal subgroup $H \unlhd G$ is called cocompact if $G/H$ is compact with respect to the quotient topology.
Note that a closed normal subgroup $H \unlhd G$ is cocompact if and only if there exist a compact subset $K \subseteq G$ such that $KH = G$ (see this thread for a proof).
My question is:
Question: Does the intersection of two cocompact closed normal subgroup of $G$ have to be cocompact as well?