The strengthened Euler characteristic-signature inequality conjecture for aspherical 4-manifolds

Let $X^{4}$ be a closed, oriented, aspherical $4$-manifold. Write $\chi(X^{4})$ for its Euler characteristic and $\sigma(X^{4})$ for its signature. Strengthened Euler characteristic-signature inequality conjecture. Then

$$\chi(X^{4})\ge 3\left|\sigma(X^{4})\right|.$$

This is a stronger proposed geography constraint than the inequality $\chi(X^{4})\ge |\sigma(X^{4})|$; it is known for surface bundles over surfaces by a result of Hamenstädt, but remains open for general closed, oriented, aspherical $4$-manifolds.