Intrinsic-bracket conjecture for degree-zero natural operations

Let ${\mathcal P}$ be an arbitrary nontrivial quadratic Koszul operad. The intrinsic bracket gives a natural homomorphism of dg operads

$$I:({\mathcal L{\it ie}},0)\to({\mathcal B}_{\mathcal P},\delta_{\mathcal P}).$$

Intrinsic-bracket conjecture. The homomorphism $I$ induces an isomorphism of operads

$$H^0({\mathcal B}_{\mathcal P})\cong{\mathcal L{\it ie}}.$$

This conjecture identifies all degree-zero natural operations with the Lie operad for every nontrivial quadratic Koszul operad; the source presents it as motivated by computational evidence and gives no proof.