Let $S^1$ be a circle.
Let $\pi: \mathbb{R}\longrightarrow S^1$ be a covering map.
Let $\rho: \pi_1(S^1)=\mathbb{Z} \longrightarrow O(n)$ be an orthogonal representation.
Let $\pi_1(S^1)$ act on $C_*(\mathbb{R})$ from right by deck transformation.
Let $\pi_1(S^1)$ act on $\mathbb{R}^n$ from left by $\rho$.
Define the $\rho$-twisted chain complex
$C_*(\mathbb{R})\otimes_{\pi_1(S^1)} \mathbb{R}^n$.
Question. Is the homology of the $\rho$-twsitec chain complex always trivial for any $\rho$?