The Hopf-Chern conjecture on Euler characteristics of negatively curved manifolds

Let $M$ be a compact Riemannian manifold of real dimension $2n$ with negative sectional curvature. Its topological Euler characteristic is denoted by $\chi(M)$.

Hopf-Chern conjecture.

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

This is a partial affirmative form of the Hopf-Chern conjecture in the setting of Kähler hyperbolic manifolds. The general conjecture for compact negatively curved manifolds remains open.