Charney–Davis flag complex conjecture

Let $S$ be a flag triangulation of a $(2k-1)$-sphere. For a finite simplicial complex $L$, define

$$\kappa(L)=\sum_{i=-1}^{\dim L}\left(-\frac12\right)^{i+1}f_i(L),$$

where $f_i(L)$ is the number of $i$-simplices and $f_{-1}(L)=1$.

Flag complex conjecture.

$$(-1)^k\kappa(S)\ge 0.$$

For nonpositively curved piecewise Euclidean cubical manifolds, this conjecture is equivalent to the Euler characteristic conjecture. The paper proves it in dimension $3$, while the general statement remains open.