Equivariant E(n)-local spectrum image conjecture

Let $G$ be a finite group. Let $\SHGEn^c$ denote the compact objects in the $E(n)$-local equivariant stable homotopy category and let $\SHG^c$ denote the compact objects in the equivariant stable homotopy category. The extension-of-scalars functor induces a continuous injective map

$$\varphi_G\colon \Spc(\SHGEn^c)\to\Spc(\SHG^c).$$

Equivariant E(n)-local spectrum image conjecture. The map $\varphi_G$ is a homeomorphism onto its image.

The preceding discussion establishes continuity and injectivity, while explicitly noting that the authors are unable to prove that the continuous bijection onto the image is a homeomorphism. Thus the topological assertion remains open in the source.