  The Goldberg conjecture states that a compact almost K\"ahler, Einstein
manifold must be K\"ahler. The analogue of this conjecture in $G_2$-geometry
says that an Einstein closed $G_2$-structure on a compact manifold must be
torsion-free, and this was proven to be true by Cleyton-Ivanov and Bryant. In
this paper, we remove the compactness assumption and prove the $G_2$-Goldberg
conjecture for noncompact manifolds. In general, we prove that there is no
closed $G_2$-structure on a noncompact manifold whose metric is complete and
has Ricci curvature pinched sufficiently close to a negative constant. We also
find restrictions on the lengths of geodesics when the metric is incomplete.
