  We prove asymptotic upper bounds for the $L^2$ Betti numbers of the locally
symmetric spaces associated to a quasi-split $U(4)$. These manifolds are
8-dimensional, and we prove bounds in degrees 2 and 3, with the behaviour in
the other degrees being well understood. In degree 3, we conjecture that these
bounds are sharp. Our main tool is the endoscopic classification of automorphic
representations of U(N) by Mok.
