Chevalley–Eilenberg embedding conjecture for perfect Lie algebras

Let $\operatorname{Lie}^{\operatorname{perf}}_{\Bbbk}\coloneqq\operatorname{Alg}_{\operatorname{Lie}}(\operatorname{Perf}_{\Bbbk})$ be the $\infty$-category of Lie algebras whose underlying $\Bbbk$-module is perfect. Let $\varepsilon\operatorname{-}\operatorname{CAlg}^{\operatorname{gr}}_{\Bbbk}$ be the $\infty$-category of commutative mixed graded $\Bbbk$-algebras. Perfect Chevalley–Eilenberg embedding conjecture. The functor

$$\operatorname{CE}^{\varepsilon}\colon\operatorname{Lie}^{\operatorname{op}}_{\Bbbk}\to\varepsilon\operatorname{-}\operatorname{CAlg}^{\operatorname{gr}}_{\Bbbk}$$

restricts to a fully faithful embedding of $\operatorname{Lie}^{\operatorname{perf}}_{\Bbbk}$. This is the cohomological analogue of the coalgebra embedding claim; the supplied text gives no resolution.