Ricci-flat tangent cone rigidity for complete Kähler manifolds

Let $M$ be a complete noncompact Kähler manifold with nonnegative Ricci curvature and Euclidean volume growth. A tangent cone at infinity is a pointed Gromov–Hausdorff limit of rescalings of $M$ at infinity, and assume that one such tangent cone is Ricci flat.

Ricci-flat tangent cone conjecture. If one tangent cone at infinity is Ricci flat, then $M$ is Ricci flat. Consequently, the tangent cone at infinity should be unique; analogous questions can be posed for local tangent cones.

The paper proves this conclusion under additional hypotheses in several corollaries. The conjectural statement removes those extra assumptions, and its local-tangent-cone analogue is likewise posed as a question.