The Lada–Markl enveloping conjecture for $L_\infty$ algebras

Let $L$ be an $L_\infty$ algebra. Write $\mathcal C(L)$ for its Chevalley–Eilenberg coalgebra, $\mathcal U L$ for Lada and Markl's universal enveloping $A_\infty$ algebra, and $B\mathcal U L$ for its bar construction. Lada–Markl's enveloping conjecture. There is a natural differential graded coalgebra quasi-isomorphism

$$\mathcal C(L) \xrightarrow{\simeq} B\mathcal U L.$$

This conjecture proposes the $L_\infty$ analogue of the classical relationship between the Chevalley–Eilenberg coalgebra and the bar construction of a universal enveloping algebra. The supplied text gives no resolution, so its status remains open.