Chevalley–Eilenberg coalgebra embedding conjecture

Let $\Bbbk$ be the base ring. Write $\operatorname{Lie}_{\Bbbk}$ for the $\infty$-category of Lie algebras, and let $\varepsilon\operatorname{-}\operatorname{cCcAlg}^{\operatorname{gr}}_{\Bbbk//\Bbbk}$ denote the $\infty$-category of cocommutative mixed graded coalgebras under and over $\Bbbk$. Let $\varepsilon\operatorname{-}\operatorname{CEcAlg}^{\operatorname{gr}}_{\Bbbk}$ be the full subcategory of coalgebras of Chevalley–Eilenberg type. Chevalley–Eilenberg coalgebra embedding conjecture. The $\infty$-functor

$$\operatorname{CE}_{\varepsilon}\colon\operatorname{Lie}_{\Bbbk}\to\varepsilon\operatorname{-}\operatorname{cCcAlg}^{\operatorname{gr}}_{\Bbbk//\Bbbk}$$

is fully faithful, with essential image exactly $\varepsilon\operatorname{-}\operatorname{CEcAlg}^{\operatorname{gr}}_{\Bbbk}$. This conjecture would characterize precisely which mixed graded cocommutative coalgebras arise by the Chevalley–Eilenberg construction; the supplied text gives no resolution.