Fully faithful enhanced Chevalley–Eilenberg group functor conjecture

Let $\operatorname{Grp}(\operatorname{Lie}_{\Bbbk})$ be the $\infty$-category of group objects in Lie algebras, and let

$$\left| - \right|^{\operatorname{t}}\colon\operatorname{Grp}(\varepsilon\operatorname{-}\operatorname{CEcAlg}^{\operatorname{gr}}_{\Bbbk})\to\operatorname{Grp}(\operatorname{cCcAlg}_{\Bbbk//\Bbbk})$$

be the Tate-realization functor, denoted in the source by

$$. **Enhanced Chevalley–Eilenberg group functor conjecture.** The functor$$

is fully faithful. The claim is presented as a reformulation of the preceding Chevalley–Eilenberg embedding conjecture using universal enveloping Hopf structures; the supplied text gives no resolution.