Fully faithful Chevalley–Eilenberg functor for representations

Fix a Lie algebra $\mathfrak{g}$ whose underlying $\Bbbk$-module is perfect. Let $\operatorname{LMod}_{\operatorname{U}(\mathfrak{g})}$ be the $\infty$-category of left modules over its universal enveloping algebra, and let $\operatorname{Mod}^{\operatorname{const}}_{\operatorname{CE}^{\varepsilon}(\mathfrak{g})}(\varepsilon\operatorname{-}\operatorname{Mod}^{\operatorname{gr}}_{\Bbbk})$ be the full subcategory of constant mixed graded Chevalley–Eilenberg modules. Representation Chevalley–Eilenberg conjecture. The functor

$$\operatorname{CE}^{\varepsilon}(\mathfrak{g};-):\operatorname{LMod}_{\operatorname{U}(\mathfrak{g})}\longrightarrow\operatorname{Mod}^{\operatorname{const}}_{\operatorname{CE}^{\varepsilon}(\mathfrak{g})}(\varepsilon\operatorname{-}\operatorname{Mod}^{\operatorname{gr}}_{\Bbbk})$$

is fully faithful. This conjectures that representations of a perfect Lie algebra can be recovered from their constant mixed graded Chevalley–Eilenberg modules; the supplied text gives no resolution.