A note on the Segal conjecture for large objects

The Segal conjecture for Cₚ (as proved by Lin and Gunawardena) asserts that the canonical map from the p-complete sphere spectrum to the Tate construction for the trivial action of Cₚ on the p-complete sphere spectrum is an isomorphism. In this article we extend the collection of spectra for which the canonical map X → XtCₚ is known to be an isomorphism to include any p-complete, bounded below spectrum whose mod p homology, viewed a module over the Steenrod algebra, is complete with respect to the maximal ideal I ⊆ A.