Telescope conjecture

Counterexample

Fix a prime $p$ and $n \ge 1$. Let $X$ be a finite $p$-local spectrum of type $n$. Let $v \colon \Sigma^d X \to X$ be a $v_n$ self-map, as given by the periodicity theorem among the Ravenel conjectures. The telescope of $v$ is the homotopy colimit $$v^{-1} X = \operatorname{hocolim}\left( X \xrightarrow{v} \Sigma^{-d} X \xrightarrow{v} \Sigma^{-2d} X \to \cdots \right).$$ The telescope conjecture asserted that the natural map $v^{-1} X \to L_{K(n)} X$ is an equivalence, where $L_{K(n)}$ is Bousfield localization at the $n$-th Morava K-theory. The conjecture is false for every prime $p$ and every height $n \ge 2$. In 2023 Burklund, Hahn, Levy and Schlank showed that for such $p$ and $n$ the map $v^{-1} X \to L_{K(n)} X$ is not an equivalence.