Koszul-operad bar-complex resolution conjecture

Let $\mathcal{P}$ be an operad and $A$ a $\mathcal{P}$-algebra. Write ${{\mathsf{B}}_{\bullet}^{\mathcal{P}}\!A}$ for the bar-complex of $A$ and $\Omega_{_{\!{\mathcal{P}}}}^1A$ for its module of Kähler differentials.

Bar-complex resolution conjecture. If $\mathcal{P}$ is a Koszul operad, then ${{\mathsf{B}}_{\bullet}^{\mathcal{P}}\!A}$ is a resolution of the left $A$-module $\Omega_{_{\!{\mathcal{P}}}}^1A$; equivalently,

$$H_i({{\mathsf{B}}_{\bullet}^{\mathcal{P}}\!A})=0,\qquad \forall i>1.$$

This generalizes the exactness of the associative bar resolution to algebras over Koszul operads. If true, it yields the stated Tor and Ext descriptions of operadic homology and cohomology; the source provides no resolution status.