Chains which are coset spaces of $tl$-groups

Let $G$ be a lattice-ordered group with a topology with which $G$ is a Hausdorff topological group and topological lattice. Let $N$ be a closed prime convex $l$-subgroup of $G$ and let $G/N$ denote the topological chain of right cosets of $N$. It is shown that if $G$ is locally compact, locally connected, or locally convex then $G/N$ is either discrete or has precisely the interval topology.