  The Segal conjecture for $C_p$ (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 $C_p$ on the $p$-complete sphere
spectrum is an isomorphism. In this article we extend the collection of spectra
for which the canonical map $X \to X^{tC_p}$ 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 $I \subseteq \mathcal{A}$.
