Counting non-uniform lattices

In [BGLM] and [GLNP] it was conjectured that if H is a simple Lie group of real rank at least 2, then the number of conjugacy classes of (arithmetic) lattices in H of covolume at most x is x(γ(H)+o(1)) log x/ log log x where γ(H) is an explicit constant computable from the (absolute) root system of H. In [BLu] we disproved this conjecture. In this paper we prove that for most groups H the conjecture is actually true if we restrict to counting only non-uniform lattices.