Aspherical manifolds with nonvanishing tautological classes

For every even integer $m\geq 2$, we construct a closed, orientable, smooth, aspherical $(2m+1)$-manifold whose fundamental group has nontrivial center and for which infinitely many tautological classes in the $\mathbb{F}_2$-cohomology of the classifying space of its group of homotopically trivial d