The eigenvalue one conjecture for finite groups

Let $G$ be a finite group, $V$ a finite-dimensional real $G$-module, and $\rho \colon G \rightarrow \operatorname{GL}(V)$ the representation afforded by $V$. Let $n \in \operatorname{GL}(V)$ be an element of finite order normalizing $\rho(G)$. The triple $(G,V,n)$ has the $E1$-property if there is some $g \in G$ such that $\rho(g)n$ has eigenvalue $1$; the group $G$ has the $E1$-property if $(G,V')$ has this property for every irreducible, non-trivial real $G$-module $V'$ of odd dimension. The eigenvalue one conjecture. Every finite group has the $E1$-property. The conjecture generalizes the original conjecture of Dekimpe, De Rock, and Penninckx. The examples preceding it include elementary abelian groups, while the extraspecial group $2_+^{1+4}$ of order $32$ provides an example that does not have the $E1$-property at the level of a particular module.