Kernel conjecture for restricted natural operations of quadratic Koszul algebras

Let ${\cal P}$ be a quadratic Koszul operad. Let ${\cal S}^*_{{\cal P}_\infty}$ be the suboperad of natural operations generated by the restricted class of operations generalizing braces, and let

$$d:{\cal S}^0_{{\cal P}_\infty}\to {\cal S}^1_{{\cal P}_\infty}$$

be its differential between degrees $0$ and $1$. Let ${\cal L{\it ie\/}}$ be the intrinsic Lie operad contained in ${\cal S}^0_{{\cal P}_\infty}$. Kernel conjecture. For each quadratic Koszul operad ${\cal P}$,

$${\cal L{\it ie\/}}\cong \operatorname{Ker}\left(d:{\cal S}^0_{{\cal P}_\infty}\to {\cal S}^1_{{\cal P}_\infty}\right).$$

This is the kernel formulation implied by the preceding generalized conjecture: it predicts that the degree-zero cocycles in the restricted natural-operations operad are exactly the intrinsic Lie operations. The supplied text gives no resolution.