  We show that if a closed hyperbolic 3-manifold has infinitely many finite
covers of bounded Heegaard genus, then it is virtually fibered. This
generalizes a theorem of Lackenby, removing restrictions needed about the
regularity of the covers. Furthermore, we can replace the assumption that the
covers have bounded Heegaard genus with the weaker hypotheses that the Heegaard
splittings for the covers have Heegaard gradient zero, and also bounded width,
in the sense of Scharlemann-Thompson thin position for Heegaard splittings.
