The geography conjecture for closed aspherical 4-manifolds

Let $X$ be a closed oriented aspherical 4-manifold. Write $\chi(X)$ for its Euler characteristic and $\sigma(X)$ for its signature. Geography conjecture. One should have

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

This is a four-dimensional consequence of Singer's conjecture and would imply the Hopf–Thurston non-negativity conjecture for Euler characteristics. The paper proves that the inequality is sharp by constructing examples with $\chi(X)=\sigma(X)=n$ for every natural number $n$, but does not state that the inequality itself is resolved in full.