conjecture is still true (again for "most" groups) if one restricts to counting non-uniform lattices. A crucial ingredient of the argument in  is the existence of towers of field extensions with bounded root discriminant which follows from the seminal work of Golod and Shafarevich on class field towers.
I plan to give an overview of these recent results and discuss some ideas beyond the proofs.
 M. Belolipetsky (with an appendix by J. Ellenberg and A.
Venkatesh), Counting maximal arithmetic subgroups, arXiv:
 M. Belolipetsky, A. Lubotzky, Class field towers and subgroup
growth, work in progress.