Hopf–Chern Euler characteristic conjecture for closed aspherical manifolds

Let $M^{2k}$ be a closed, aspherical manifold of dimension $2k$, and let $c7(M^{2k})$ denote its Euler characteristic.

Euler characteristic conjecture.

$$(-1)^k\chi(M^{2k})\ge 0.$$

This conjecture is attributed to Hopf in the nonpositively curved Riemannian case and was suggested more generally by Thurston. It is known in several special cases, including dimensions $2$ and $4$ for nonpositively curved Riemannian manifolds, but remains open in general.