The 3-dimensional manifold Charney–Davis conjecture

Let $\Delta$ be a flag 3-manifold, and let $\beta_1(\Delta)$ denote its first Betti number. 3-dimensional manifold Charney–Davis conjecture.

$$\gamma_2(\Delta)\geq 16\beta_1(\Delta).$$

The conjecture extends the Davis–Okun inequality from flag 3-spheres to flag 3-manifolds and is motivated by computed examples relating minimal $\gamma_2$ to first Betti number. Its general validity remains open.