Type II classification conjecture for gravitational instantons

Let $(X,g)$ be a Type II gravitational instanton, meaning a complete Ricci-flat 4-manifold with finite energy whose self-dual Weyl tensor has exactly two distinct eigenvalues everywhere. Let $\operatorname{ALF}$ and $\operatorname{AF}$ denote the corresponding asymptotic-fibered and asymptotically-flat model classes, and let the Eguchi–Hanson metric be equipped with the reversed orientation.

Type II classification conjecture. All Type II gravitational instantons are toric and must be ALF or AF, except the Eguchi-Hanson metric with the reversed orientation.

Known classification results restrict Type II gravitational instantons with non-Euclidean volume growth to several model families, while the conjecture asserts the stronger global toric and ALF/AF classification, with the stated Eguchi–Hanson exception. The conjecture remains open.