  By assuming the endoscopic classification of automorphic representations on
inner forms of unitary groups, which is currently work in progress by Kaletha,
Minguez, Shin, and White, we bound the growth of cohomology in congruence
towers of locally symmetric spaces associated to $U(n,1)$. In the case of
lattices arising from Hermitian forms, we conjecture that the growth exponents
we obtain are sharp in all degrees.
