Charney–Davis conjecture for flag odd-dimensional spheres

Let $k\geq 2$. A flag sphere is a simplicial sphere with no missing faces of dimension larger than one, and $g_0,g_1,\ldots,g_k$ are its $g$-numbers. Charney–Davis conjecture. Every flag $(2k-1)$-sphere satisfies

$$g_k-g_{k-1}+\dots+(-1)^kg_0\geq 0.$$

This is known for $k=2$ but remains open in higher dimensions; it is motivated by the Hopf conjecture in Riemannian geometry.