Rognes' localization conjecture for Johnson–Wilson spectra

Let $p$ be a prime, and let $BP\langle n\rangle^{\scriptscriptstyle\wedge}_{p}$ denote the $p$-completed connective Johnson–Wilson spectrum, with $E(n)^{\scriptscriptstyle\wedge}_{p}$ its $p$-completed periodic version. Consider the transfer map and canonical map

$$K(BP\langle n-1\rangle^{\scriptscriptstyle\wedge}_{p})\longrightarrow K(BP\langle n\rangle^{\scriptscriptstyle\wedge}_{p})\longrightarrow K(E(n)^{\scriptscriptstyle\wedge}_{p}).$$

Rognes' localization conjecture. These maps should fit into a cofiber sequence in the stable category

$$K(BP\langle n-1\rangle^{\scriptscriptstyle\wedge}_{p})\longrightarrow K(BP\langle n\rangle^{\scriptscriptstyle\wedge}_{p})\longrightarrow K(E(n)^{\scriptscriptstyle\wedge}_{p})\longrightarrow\Sigma K(BP\langle n-1\rangle^{\scriptscriptstyle\wedge}_{p}).$$

This conjecture is part of a program relating the chromatic tower to algebraic $K$-theory of Morava $E$-theory and Johnson–Wilson spectra. The source presents it as a conjectural localization sequence and does not state that it is resolved.