The conformal Ricci-flatness conjecture for one-ended manifolds

Let $(M,g)$ be a complete, noncompact manifold with only one end: outside a compact set it is diffeomorphic to a spherical shell $S^{n-1}\times\mathbb{R}$. Let $g_0$ denote the Euclidean metric and let $\phi$ be a regular function. Conformal Ricci-flatness conjecture. There is a metric $g'$ conformally equivalent to $g$ such that its Ricci curvature vanishes on the complement of some compact set and

$$g'=e^{\phi}g_0.$$

The conjecture asks whether the geometry of a one-ended complete manifold can be made Euclidean-conformal and Ricci-flat near infinity. The source leaves both the precise domain of regularity of $\phi$ and the conjecture itself unresolved.