The eigenvalue one conjecture for odd-dimensional faithful irreducible representations

Let $G$ be a non-trivial finite group and let

$$\rho\colon G\rightarrow \operatorname{GL}_d(\mathbb{Z})$$

be a representation. Write $\rho^{\mathbb{R}}$ for the representation obtained by extending scalars to $\mathbb{R}$, and let $N_{\operatorname{GL}_d(\mathbb{Z})}(\rho(G))$ denote the normalizer of $\rho(G)$ in $\operatorname{GL}_d(\mathbb{Z})$. Assume that $\rho^{\mathbb{R}}$ is faithful and irreducible, and that $d$ is odd. Eigenvalue one conjecture. For every $n\in N_{\operatorname{GL}_d(\mathbb{Z})}(\rho(G))$, there exists $g\in G$ such that $\rho(g)n$ has eigenvalue $1$. This conjecture gives a sufficient eigenvalue condition used to establish the $R_\infty$ property for closed flat manifolds; its resolution status is not specified in the supplied text.