The Bousfield Class of the Infinite Wedge of Telescopes of Finite Spectra

Question

The spectrum $T(n)$ which is the telescope of a finite spectrum of type n along its self-map, has a unique Bousfield class $\langle T(n)\rangle$ which only depends on $n$. It is also known, from Ravenel's paper "Localization with respect to certain periodic homology theories" that $\langle S\rangle=\langle T(0)\rangle\vee\langle T(1)\rangle\vee\ldots\vee\langle T(n-1)\rangle\vee\langle F(n)\rangle$ where $F(n)$ is a finite spectrum of type n. Is it known whether or not $\langle S\rangle=\bigvee_n\langle T(n)\rangle$?

In the same vein, is it known whether or not $\bigvee_n\langle K(n)\rangle=\langle S\rangle$?

Answer 1

The spectrum $H\mathbf{F}_p$ is an acyclic for both $\bigvee \langle T(n) \rangle$ and $\bigvee \langle K(n) \rangle$. Therefore neither of these wedges equals $\langle S \rangle$.