The Charney–Davis conjecture for non-positively curved piecewise Euclidean manifolds

Let $M^d$ be a compact, closed, $d$-dimensional manifold with a piecewise Euclidean structure, and suppose that it is non-positively curved in the sense of Gromov. Let $\chi(M^d)$ denote its Euler characteristic. Charney–Davis conjecture. If $d$ is even, then

$$(-1)^{\frac{d}{2}}\chi(M^d)\geq 0.$$

This is a combinatorial analogue of Hopf's conjecture on the Euler characteristic of closed non-positively curved manifolds. The source presents it as open in general, although it is known in dimensions $2$ and $4$ in the corresponding smooth setting.