The algebraic chromatic splitting conjecture

Assume $p$ is sufficiently large with respect to $n$. Let $E$ be the height-$n$ theory in the source, let $I_{n-1}$ be the corresponding invariant ideal, and let $L_{n-1}$ and $L_{K(n)}$ denote the stated localization functors. For $M$ in the thick subcategory $\operatorname{Thick}(E_*/I_{n-1})$, let $\iota_M$ and $\zeta_M$ be the maps constructed in the surrounding discussion. The algebraic chromatic splitting conjecture. For every

$$M\in\operatorname{Thick}(E_*/I_{n-1}),$$

there is an equivalence

$$L_{n-1}M\oplus\Sigma^{-1}L_{n-1}M\simeq L_{n-1}L_{K(n)}M,$$

induced by $\iota_M$ and $\zeta_M$.

This is an algebraic analogue of the chromatic splitting conjecture. The construction of the class $\zeta$ is described as work in progress, and the source gives no resolution status for the conjecture.