Linear and smooth oriented equivalence conjecture for finite-group representations

Let $n$ be a positive integer, let $\Gamma$ be a finite group, and let $\rho,\rho':\Gamma\to\operatorname{O}(n)$ be homomorphisms. Two such homomorphisms are $H$-equivalent if they are conjugate by an element of $H$. Here $\operatorname{SO}(n)$ is the orientation-preserving subgroup of $\operatorname{O}(n)$, and $\operatorname{Diff}^{+}(S^{n-1})$ is the group of orientation-preserving diffeomorphisms of $S^{n-1}$. Linear and smooth oriented equivalence conjecture. If $\rho$ and $\rho'$ are $\operatorname{O}(n)$-equivalent and $\operatorname{Diff}^{+}(S^{n-1})$-equivalent, then they are $\operatorname{SO}(n)$-equivalent. The conjecture concerns whether simultaneous linear and orientation-preserving smooth conjugacy can always be realized by an orientation-preserving orthogonal conjugacy; the paper proves it for $n\leq 5$, while the general case remains open.