The Mahowald-Rezk telescope conjecture for fp-spectra

Let $X$ be a spectrum of fp-type $n$, and let $L_n^f$ and $L_n$ denote, respectively, finite and ordinary chromatic localization at height $n$. Mahowald-Rezk conjecture. $X$ satisfies the telescope conjecture in the sense that

$$L_n^f X \longrightarrow L_n X$$

is an equivalence. The Hahn-Wilson conjecture would imply this statement because $\mathrm{BP} \langle n \rangle$ satisfies the telescope conjecture as an $\mathrm{MU}$-module. However, the source reports counterexamples to the finiteness condition underlying the Mahowald-Rezk conjecture, so this conjecture is disproved.