The twisted Alexander polynomial detection conjecture for fibered 3-manifolds

Let $N$ be a $3$--manifold and let $\phi\in H^1(N)$ be a non-trivial class. For an epimorphism $\alpha:\pi_1(N)\to G$ onto a finite group, let $\Delta^{\alpha}_{N,\phi}\in\mathbb Z[t^{\pm1}]$ be the associated twisted Alexander polynomial, let $\|\phi\|_T$ denote the Thurston norm, and let $\operatorname{div}\phi_G$ denote the divisibility of the induced class on the finite cover associated to $\alpha$.

Twisted Alexander polynomial detection conjecture. If, for every epimorphism $\alpha:\pi_1(N)\to G$ onto a finite group, $\Delta^{\alpha}_{N,\phi}$ is monic and

$$\deg\Delta^{\alpha}_{N,\phi}=|G|\,\|\phi\|_T+2\operatorname{div}\phi_G,$$

then $(N,\phi)$ fibers over $S^1$.

The paper explains that twisted Alexander polynomials provide effective obstructions to fiberability and that the conjecture would characterize fibered classes through all finite-group quotients. The conjecture is presented as an aim of the paper and is not resolved there.