Gromov positive scalar curvature conjecture and rationally inessential macroscopically large manifolds

We give the first examples of rationally inessential but macroscopically large manifolds. Our manifolds are counterexamples to the Dranishnikov rationality conjecture. For some of them, we prove that they do not admit a metric of positive scalar curvature (PSC), and thus satisfy the Gromov PSC conjecture. Fundamental groups of our manifolds are finite index subgroups of right-angled Coxeter groups. The construction uses small covers of convex polyhedrons (or alternatively Davis complexes) and surgery.