A counterexample to a $BP$-analogue of the chromatic splitting conjecture

We prove that, if $n\ge 2$, the $E(n-1)_\ast$-localization of the $K(n)_\ast$-localization map $BP_p\to L_{K(n)}BP$ is not a split monomorphism in the stable category by exhibiting spectra $Z$ for which the map $\pi _\ast(L_{n-1}(BP_p)\wedge Z)\to\pi _\ast(L_{n-1}(L_{K(n)}BP)\wedge Z)$ is not injective. If $p\ge \max\{\frac{1}{2}(n^2-2n +2), n+1\}$ and $n\geq 3$, we show that $Z$ may be taken to be a two-cell complex in the sense of $E(n-1)_\ast$-local homotopy theory. The question of whether the map $L_{n-1}(BP_p)\to L_{n-1}L_{K(n)}BP$ splits was asked by Hovey and is in some sense a $BP$-analogue of Hopkins' chromatic splitting conjecture.