Hamilton's flatness conjecture for Ricci-pinched noncompact three-manifolds

Let $(M^3,g)$ be a complete, non-compact Riemannian manifold. Say that $(M^3,g)$ is Ricci pinched when it satisfies the Ricci-pinching condition used in Hamilton's conjecture. Hamilton's conjecture. If $(M^3,g)$ is Ricci pinched, then $(M^3,g)$ is flat.

Hamilton's conjecture predicts rigidity of complete noncompact Ricci-pinched three-manifolds. The supplied text gives no resolution or further hypotheses, so its status is left open.