It is shown that all torsion-free cocompact lattices in any SL$(n,\mathbb{C})$  produce the desired integrality of the trace (under a very weak Spin$^c$-assumption) in my thesis. Very few, if any, of these groups  are known to satisfy Baum-Connes for $n\ge 3$. (I tend to think none of these are known for $n\ge 3$, but am not familiar enough with V.Lafforgues work.)

The Kadison-Kaplansky-conjecture, in turn, was known much earlier for these groups; this goes back to Hyman Bass.

Maybe one can say, therefore, that trace integrality is rather a problem in geometry than in representation theory.

You can read the thesis here:
http://arxiv.org/abs/math/0612023