Let $E = E_n$ be the $n$-th Morava $E$-theory and let $\{ M_{I} \}$ be a tower of generalised Moore spectra. Then (see this previous question) there is a Milnor exact sequence

$$0 \to \varprojlim_I {}^1 (E \wedge M_I)_{\ast+1}(X) \to E_*^\vee X \to \varprojlim_I (E \wedge M_I)_*X \to 0. $$

Here we write $E_*^\vee X$ for $\pi_*L_{K(n)}(E \wedge X)$.

The functor $\mathcal{K}_*X := \varprojlim_I (E \wedge M_I)_*X$ is sometimes called the Morava module of $X$, and seems to have first been introduced by Hopkins, Mahowald and Sadofsky whilst studying the $K(n)$-local Picard group.

In Hopkins-Mahowald-Sadofksy, under the condition that $\mathcal{K}_*X$ is finitely-generated, the authors show that the $E_2$-term of the $K(n)$-local $E_n$-based Adams spectral sequence can be given by $H_c^\ast(\mathbb{G}_n,\mathcal{K}_*X)$, where $\mathbb{G}_n$ is the $n$-th (extended) Morava stabilizer group.

It is known that the $\varprojlim{}^1$ term of the Milnor exact sequence vanishes under suitable conditions; for example if $E_*^\vee X$ is finitely-generated, or pro-free, as an $E_*$-module. In these cases then $E^\vee_*X \simeq \mathcal{K}_*X$. I suspect that the above spectral sequence should have $E_2$-term $H_c^\ast(\mathbb{G}_n,E^\vee_*X)$. This leads to the following question:

If $\mathcal{K}_*X$ is finitely-generated is there an isomorphism $\mathcal{K}_*X \simeq E^\vee_*X$?

By the comments above it would suffice to show that $E^\vee_* X$ is finitely-generated. Hovey-Strickland Proposition 8.6 gives a number of equivalent conditions, including that $X$ is $K(n)$-locally dualisable or that $K_*X$ is finite.