The vanishing Alexander polynomial conjecture for 3-manifolds

Let $b\geq4$ be an integer, and let $M$ be an orientable connected closed $3$--manifold with $b_1(M)=b$. A primitive cohomology class is a class $\psi\in H^1(M;{\mathbb Z})$ that is primitive in the integral cohomology lattice. Let $\Delta_M^\psi(t)$ denote the Alexander polynomial associated with $\psi$.

Vanishing Alexander polynomial conjecture. There exists some primitive cohomology class $\psi\in H^1(M;{\mathbb Z})$ such that the Alexander polynomial $\Delta_M^\psi(t)$ vanishes.

The conjecture is motivated by the difficulty of extending the paper's partial examples for first Betti number at least $4$ to the full virtual Euler class one conjecture. It is stated as an open question about Alexander polynomials for closed $3$--manifolds.