Let $R$ be an $L_n$-local ring spectrum. Then one knows that the Tate construction $R^{tC_p}$ (with respect to the trivial $C_p$-action on $R$) is $L_{n-1}$-local; this "blueshift" result is due to Hovey-Sadofsky and Greenlees-Sadofsky, and has been generalized to the telescopic setting by Kuhn.
I am curious to what extent there is an analog of this result when $R$ is instead connective. For example, suppose $R$ has the property that its $K(i)$-localization vanishes for $i \geq n$ (not including $\infty$, so for instance the connective cover of an $L_n$-local ring spectrum is allowed). Does this imply that the $K(i)$-localization of $R^{tC_p}$ vanishes for $i \geq n-1$?