Graph manifolds as ends of negatively curved Riemannian manifolds

Let $M$ be a graph manifold such that each piece of its JSJ decomposition has the ${\mathbb{H}}^2\times \mathbb{R}$ geometry. Assume that the pieces are glued by isometries. Then, there exists a complete Riemannian metric on $\Bbb R \times M$whichisan\mathrm{"}eventuallywarpedcuspmetric\mathrm{"}withthesectionalcurvature$K$satisfying$-1 \le K <0$. A theorem by Ontaneda then implies that $M$ appears as an end of a 4-dimensional, complete, non-compact Riemannian manifold of finite volume with sectional curvature $K$ satisfying $-1\le K<0$.