Thurston's virtually fibered conjecture for three-manifolds

A fibered three-manifold is a three-manifold admitting a submersion $p:M\to S^1$; equivalently, it is a mapping torus

$$M=\frac{F\times I}{(x,0)\sim(\psi(x),1)}$$

for a surface $F$ and an orientation-preserving isomorphism $\psi:F\to F$. A three-manifold is virtually fibered if it has a finite cover that is fibered. Thurston's virtually fibered conjecture. Let $M$ be a compact orientable irreducible three-manifold whose fundamental group is infinite and contains no non-peripheral $\mathbb{Z}\times\mathbb{Z}$ subgroup. Suppose that every boundary component of $M$ is a torus. Then $M$ is virtually fibered. This conjecture predicts virtual fibering for a broad class of irreducible three-manifolds; the source gives no evidence of a resolution.