The Milnor–Moore conjecture for $A_\infty$ algebras

Let $A$ be an $A_\infty$ algebra over a characteristic-zero field with a cocommutative, conilpotent coproduct $\Delta$ that is a strict $A_\infty$ morphism

$$A\to A^{\otimes 2}.$$

Let $\overline\Delta$ denote the reduced coproduct and set $L=\operatorname{Ker}(\overline\Delta)=\mathcal P_*(A)$, the primitives of $A$. Milnor–Moore conjecture. The primitives $L$ form an $L_\infty$ algebra, and the inclusion $L\hookrightarrow A$ extends to an isomorphism of $A_\infty$ algebras

$$UL\xrightarrow{\cong} A$$

that respects the Hopf structure. This is the proposed infinity analogue of the Cartier–Milnor–Moore theorem, characterizing suitable cocommutative Hopf-type objects by their primitives. The supplied text gives no resolution, so the conjecture remains open.