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).