Linearly controlled asymptotic dimension of the fundamental group of a graph-manifold

We prove the estimate $\ell\text{-asdim}\,\pi_1(M)\leq7$ for the linearly controlled asymptotic dimension of the fundamental group of any 3-dimensional graph-manifold $M$. As applications, we show that the universal cover $\widetilde M$ of $M$ is an absolute Lipschitz retract and admits a quasisymmetric embedding into the product of 8 metric trees.