Quasi-isometry of pairs: surfaces in graph manifolds

We show there exists a closed graph manifold $N$ and infinitely many non-separable, horizontal surfaces $\{S_{n} \to N\}_{n \in \mathbb{N}}$ such that there does not exist a quasi-isometry $\pi_1(N) \to \pi_1(N)$ taking $\pi_1(S_{n})$ to $\pi_1(S_{m})$ within a finite Hausdorff distance when $n \neq m$.