The periodic-representation detection conjecture for nonfibered knots

Let $k$ be a knot with augmented group system ${\cal G}=(G,\epsilon,x)$. Let $r$ denote the period parameter, let $S_N$ be the symmetric group on $N$ letters, and let $\Phi_{S_N}({\cal G})$ denote the set of periodic representations of the augmented group system into $S_N$. For such a representation $\rho$, write $D_{\rho,r}(s)$ for the associated polynomial.

Periodic-representation detection conjecture. The knot $k$ is nonfibered if and only if there exist an integer $N>0$ and a periodic representation $\rho\in\Phi_{S_N}({\cal G})$ such that

$$D_{\rho,r}(s)=0.$$

This gives an algebraic detection criterion for fiberedness using twisted Alexander-type invariants: the conjecture asserts that every nonfibered knot admits a periodic representation for which the specified polynomial vanishes, while fibered knots admit no such representation.