Generalized Chern Conjecture for closed oriented aspherical manifolds

Let $M$ be a closed oriented aspherical manifold. Its tangent bundle is denoted by $TM$, and a flat structure on $TM$ means that it is induced by a representation of the fundamental group of $M$. Generalized Chern Conjecture. If $TM$ admits a flat structure, then

$$\chi(M)=0.$$

This conjecture concerns the vanishing of the Euler characteristic for closed oriented aspherical manifolds whose tangent bundles are flat. The source presents the result as an important application of Milnor–Wood inequalities, but the supplied material gives no definitive resolution status.