Asymptotic dimension of graph manifold groups

Question

Does every non-geometric graph manifold have fundamental group of asymptotic dimension 3?

This is affirmed in http://arxiv.org/abs/0909.1098 for closed graph manifolds, but I am interested in non-closed graph manifolds as well. Notice that the asymptotic dimension of such groups is always at least 2 (obvious) and at most 3 (by a result of Bell and Dranishnikov).

Answer 1

I should have seen much earlier that it's 2. In fact, every non-closed graph manifold has a finite sheeted cover that fibers over the circle, see here. The fundamental group of such cover is an HNN extension of a free group, so that it has asymptotic dimension at most 2 by the main result of this paper.