  We study the integral Hodge conjecture in complex codimension $2$ and $3$ for
approximations to the classifying space of groups of type A. In degree two, we
prove a conjecture of Ben Antieau, extending his two counterexamples to a
general family of groups. In particular, when reduced to a finite field, these
spaces extend Antieau's counterexamples to the integral $\ell$-adic Tate
conjecture from the cases $\ell = 2, 3$ to all primes $\ell$.
