Virtual fibering conjecture for taut sutured manifolds

Let $(M,\gamma)$ be a taut sutured manifold, meaning a sutured manifold satisfying the tautness condition, and suppose that $\chi(M)<0$. A finite-sheeted cover is a covering

$$(\tilde{M},\tilde{\gamma})\to(M,\gamma)$$

with finitely many sheets. The virtual fibering conjecture for taut sutured manifolds. There is a finite-sheeted cover $(\tilde{M},\tilde{\gamma})\to(M,\gamma)$ such that $(\tilde{M},\tilde{\gamma})$ has a depth one taut oriented foliation. This is presented as a natural analogue of the virtual fibering conjecture for 3-manifolds; its status is not resolved in the supplied text.