Kotschick–Löh's domination conjecture by hyperbolic manifolds

Let $M^n$ be an oriented closed manifold, where $n\ge 2$. A manifold is hyperbolic if it admits a Riemannian metric of constant negative curvature. Kotschick–Löh's conjecture. Every $n$-dimensional oriented closed manifold can be dominated by a hyperbolic manifold. This conjecture concerns domination of arbitrary closed manifolds by manifolds with constant negative curvature and is related to the construction of URC-manifolds in the paper.