Quasi-isometry conjecture for separable horizontal surfaces in graph manifolds

Let $N$ and $M$ be two closed graph manifolds. Let $S \looparrowright N$ and $S' \looparrowright M$ be two separable, horizontal surfaces. Quasi-isometry conjecture. There is a quasi-isometry from $\pi_1(N)$ to $\pi_1(M)$ mapping $\pi_1(S)$ to $\pi_1(S')$ within a finite Hausdorff distance. This conjecture proposes that separable horizontal surfaces in closed graph manifolds cannot be distinguished by the quasi-isometry type of the associated pairs; the paper establishes that the analogous assertion fails for non-separable horizontal surfaces, while no counterexample is known in the separable case.