  Suppose that $(X, g)$ is a conformally compact $(n+1)$-dimensional manifold
that is hyperbolic at infinity in the sense that outside of a compact set $K
\subset X$ the sectional curvatures of $g$ are identically equal to minus one.
We prove that the counting function for the resolvent resonances has maximal
order of growth $(n+1)$ generically for such manifolds.
