telescopic complexity $n$.

To make sense of this definition, one needs to prove that $\mathcal{T}_X$ is indeed a thick subcategory, and this is proven in [@MR2079370]. One way to phrase the red-shift conjecture is, therefore, as follows.

\[conj\] If $R$ is a commutative ring spectrum with telescopic complexity $n$, then $K(R)$ is a commutative ring spectrum with telescopic complexity $n+1$.

Note that the version of the conjecture due to C. Ausoni and J. Rognes states that if $R$ is a “suitably finite" $K(n)$-local commutative ring spectrum then for some $V$ in $\mathfrak{S}_{n}$, the map (\[eq red shift\]) induces an isomorphism in sufficiently high degrees. Since this statement is conjectural, it is possible that it may need to be reformulated for it to be true. We therefore see no issue examining the validity of statement Conjecture \[conj\], which clearly applies in our case of interest and only slightly differs from the original conjecture of C. Ausoni and J.Rognes.

In the example of interest, we know that $$S/p \wedge \tau_{\ge 0}L_{K(1)}S \longrightarrow v_1^{-1} S/p \wedge \tau_{\ge 0}L_{K(1)}S$$ induces an isomorphism in sufficiently high degrees. Since every object in $\mathfrak{S}_{0}$ can be constructed as a finite cell $S/p$-module, it suffices to check telescopic 