Finite-energy versus finite-topology conjecture for complete Ricci-flat 4-manifolds

Let $(X,g)$ be a complete Ricci-flat 4-manifold. Its energy is the curvature integral

$$\int_X |\operatorname{Rm}_g|^2.$$

Say that $X$ has finite topological type if it is diffeomorphic to the interior of a compact manifold with boundary.

Finite-energy versus finite-topology conjecture. The manifold $(X,g)$ has finite energy if and only if $X$ has finite topological type.

This conjecture links an analytic finiteness condition on the curvature to a topological finiteness condition. It is stated as a conjecture for general complete Ricci-flat 4-manifolds, beyond the better-understood hyperkähler setting, and remains open.