The twisted Iwasawa detection conjecture for non-fibered knots

Let $K$ be a knot, let $p$ be a fixed prime number, and let $\pi$ be its knot group. For a representation $\rho:\pi\to \operatorname{GL}_N O_{\mathfrak{p}}$, let $\mu_1$ denote the associated twisted Iwasawa invariant when it is defined. Twisted Iwasawa detection conjecture. If $K$ is non-fibered, then there exists some representation $\rho:\pi\to \operatorname{GL}_N O_{\mathfrak{p}}$ such that $\mu_1$ is defined and $\mu_1>0$. The preceding discussion explains why a representation with positive $\mu_1$ would detect non-fiberedness; whether such a representation always exists is left open.