Hopf conjecture on the Euler characteristic of nonpositively curved manifolds

Let $(X,g)$ be a closed $2n$-dimensional Riemannian manifold with sectional curvature $\mathrm{sec}_{g}$. Write $\chi(X)$ for its Euler characteristic. Hopf conjecture.

$$(-1)^n\chi(X)>0\quad\text{if }\mathrm{sec}_{g}<0,$$

and

$$(-1)^n\chi(X)\geq 0\quad\text{if }\mathrm{sec}_{g}\leq 0.$$

This conjecture predicts a curvature-controlled sign for the Euler characteristic and is a central problem in global differential geometry. Its status in the stated generality is not resolved here.