Quasiregular curves and cohomology

Let $N$ be a closed, connected, and oriented Riemannian manifold, which admits a quasiregular $\omega$-curve $\mathbb{R}^n \to N$ with infinite energy. We prove that, if the de Rham class of $\omega$ is non-zero and belongs to a so-called K\"unneth ideal, then there exists a non-trivial graded algebra homomorphism $H_{\mathrm{dR}}^*(N) \to \bigwedge^* \mathbb{R}^n$ from the de Rham algebra $H_{\mathrm{dR}}^*(N)$ of $N$ to the exterior algebra $\bigwedge^* \mathbb{R}^n$. As an application, we give examples of pairs $(N,\omega)$, where $N$ is a closed manifold and $\omega$ is a closed $n$-form for $n<\dim N$, for which every quasiregular $\omega$-curve $\mathbb{R}^n \to N$ is constant.