The Koszul propérade bar-to-levelled composition quasi-isomorphism

Let $\mathcal{P}$ be a Koszul propérade. Consider the composition

$$\mathcal{P}^{\scriptstyle \textrm{!`}}\to\bar{\mathcal{B}}(\mathcal{P})\to\bar{\mathcal{N}}(\mathcal{P}).$$

Koszul bar-to-levelled conjecture. The composition

$$\mathcal{P}^{\scriptstyle \textrm{!`}}\to\bar{\mathcal{B}}(\mathcal{P})\to\bar{\mathcal{N}}(\mathcal{P})$$

is a quasi-isomorphism.

The claim links the Koszul dual propérade with the bar and levelled constructions. The supplied text gives no resolution status; its relationship with the preceding criterion suggests that it may be intended as a consequence, but no proof or status is stated in the candidate span.