Given an aspherical 4-dimensional closed manifold $M$ with fundamental group $\mathbb{Z}^4$, it is homotopy-equivalent to $T^4 = S^1 \times \ldots S^1$, the 4-dimensional torus.

**Question 1:** Since I am no expert and could not dig out a reference I would be interested if it is open/known that under the circumstances above, $M$ must be homeomorphic (diffeomorphic) to $T^4$?

**Question 2:** Can one say more if one knows that $M$ is smoothly covered by $\mathbb{R} \times T^3$?

*Remark:* in all other dimensions it seems to be true (due to Wall et al.) that for dimensions $n \geq 5$ the manifold $M$ is finitely coverey by a manifold diffeomorphic to $T^n$ and for $n \leq 3$ it is even diffeomorphic to $T^3$.