Hopkins' algebraic chromatic splitting conjecture

Let $E_*$ be the coefficient ring of Morava $E$-theory at height $n$, let $E_*E$ be its cooperations, and let $v_0=p,v_1,\ldots,v_{n-1}$ denote the usual periodicity elements. For $i\geq 1$, consider the inverse systems of $E_*E$-comodules $E_*/(p,\ldots,v_{n-2},v_{n-1}^i)$ and their derived limits. Let $\iota\colon S^0\to L_{K(n)}S^0$ be the unit map and let $\zeta\colon S^{-1}\to L_{K(n)}S^0$ be the class constructed from the determinant fiber sequence. Hopkins' algebraic chromatic splitting conjecture. For all heights $n\geq 2$ and sufficiently large primes $p$, there are isomorphisms of $E_*E$-comodules

$$\varprojlim_{E_*E}^{s}E_*/(p,\ldots,v_{n-2},v_{n-1}^{i})\cong\begin{cases} E_*/(p,\ldots,v_{n-2}) & s=0,\\ v_{n-1}^{-1}E_*/(p,\ldots,v_{n-2}) & s=1,\\ 0 & \mathrm{otherwise.}\end{cases}$$

Moreover, these isomorphisms should be topologically realized by $\iota$ and $\zeta$, respectively. This algebraic form predicts the two nonzero derived-limit contributions corresponding to the unit and the degree $-1$ class in the $K(n)$-local sphere. It is motivated by height-two computations and is recorded in the cited literature; the source presents it as a conjecture and does not state that it has been resolved in the stated generality.