  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 $x^{(\gamma(H)+o(1))\log x/\log\log
x}$ where $\gamma(H)$ 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.
