Chern–Hopf–Thurston conjecture on Euler characteristics of aspherical manifolds

Let $M$ be a closed real $2n$-manifold. It is aspherical when its universal covering is contractible. Chern–Hopf–Thurston conjecture. If $M$ is aspherical, then it satisfies

$$(-1)^n\chi(M)\geq 0.$$

This generalizes the sign prediction for manifolds of non-positive sectional curvature. The conjecture is known in low dimensions but remains widely open for $n\geq 3$.