The dimension bound for compact manifolds with standard lattice actions

Let $M$ be a compact manifold with

$$\pi_{1}(M)={\mathbb Z}^{n}.$$

Suppose that $\Gamma$ acts on $M$ and that the induced action on $\pi_{1}(M)$ is the standard action.

Dimension bound conjecture. The dimension of $M$ must be at least $n$.

This would rule out space-filling curves in the setting of continuous quotient maps for lattice actions. The source presents it as a desired consequence and gives no resolution, so its status remains open.