Gromov–Lück inequality for closed aspherical 4-manifolds

Let $X$ be a closed, oriented, aspherical $4$-manifold, and let $\tau(X)$ denote its signature. Gromov–Lück inequality. One has

$$\chi(X)\geq |\tau(X)|.$$

This inequality is a refined form of the Hopf problem on the sign of the Euler characteristic of closed aspherical $4$-manifolds, motivated by the Singer conjecture on the vanishing of $L^2$-Betti numbers. Its status is not resolved in the supplied source context.