Fix a prime $p$. We let $C$ be the completion of the algebraic closure of $\mathbb{Q}_p$, and let $\mathcal{O}_C$ be its ring of integers. For a $p$-complete $K(\mathcal{O}_C;\mathbb{Z}_p)$-module $M$, there is the unit map $$ M \to L_{K(1)}M. $$ If $M$ is the p-completion of the algebraic K theory of a smooth projective variety $M$ over $C$, by Quillen-Lichtenbaum conjecture, the unit map $$ K^{alg}(X)_{\hat{p}} \to L_{K(1)}K^{alg}(X) $$ is $n$-connetced for some natural number $n$. Can we extend this story to general $M$? That is, the unit map $ M \to L_{K(1)}M $ is $n$-connetced for some natural number $n$?

Fix a prime $p$. We let $C$ be the completion of the algebraic closure of $\mathbb{Q}_p$, and let $\mathcal{O}_C$ be its ring of integers. For a $p$-complete $K(\mathcal{O}_C;\mathbb{Z}_p)$-module $M$, there is the unit map $$ M \to L_{K(1)}M. $$ If $M$ is the p-completion of the algebraic K theory of a smooth projective variety $M$ over $C$, by Quillen-Lichtenbaum conjecture, the unit map $$ K^{alg}(X)_{\hat{p}} \to L_{K(1)}K^{alg}(X) $$ is $n$-connected for some natural number $n$. Can we extend this story to general $M$? That is, the unit map $ M \to L_{K(1)}M $ is $n$-connected for some natural number $n$?