On the non-existence of certain branched covers

We prove that while there are maps ${\text{bT}}^4\to {\mathrm{\#}}^3({\text{bS}}^2\times {\text{bS}}^2)$ of arbitrarily large degree, there is no branched cover from $4$-torus to ${\mathrm{\#}}^3({\text{bS}}^2\times {\text{bS}}^2)$. More generally, we obtain that, as long as $N$ satisfies a suitable cohomological condition, any ${\pi }_1$-surjective branched cover ${\text{bT}}^n\to N$ is a homeomorphism.