Conjecture on completion and localization for commutative localized quotients

Let $R$ be a ring spectrum and let $E$ be a commutative localized quotient of $R$. For a finite cell $R$-module $M$, let $\hat{\mathrm{L}}^R_E M$ denote the completed $E$-localization and $\operatorname{L}^R_E M$ the $E$-localization. Completion–localization conjecture. The conclusion of Theorem~ holds when $E$ is any commutative localized quotient of $R$; in particular, for every finite cell $R$-module $M$,

$$\pi_*\hat{\mathrm{L}}^R_E M=M_*[X^{-1}]\sphat_{I_*}=R_*[X^{-1}]\sphat_{I_*}\oTimes{R_*}M_*.$$

If the regular sequence generating $I_*$ is finite, then the natural map $M\longrightarrow\hat{\mathrm{L}}^R_E M$ is an $E$-equivalence, so $\operatorname{L}^R_E M\simeq\hat{\mathrm{L}}^R_E M$.