Bounded Virtual Bundle Conjecture for finite-volume hyperbolic 3-manifolds

Let $N$ be a compact, connected, ${\mathbf{P}^2}$-irreducible $3$-manifold whose boundary is a non-empty collection of incompressible tori and Klein bottles. Assume that every subgroup of $\pi_1(N)$ isomorphic to $\mathbf{Z}\oplus\mathbf{Z}$ is peripheral, and that $N$ is not an $I$-bundle over a torus or Klein bottle. Bounded Virtual Bundle Conjecture. Some finite-sheeted covering space of $N$ is a surface bundle over $S^1$. The stated hypotheses correspond, by the source's discussion of hyperbolization, to the finite-volume hyperbolic setting; the source gives no resolution.