The distributive-law conjecture for Hopf operads generated by primitive nilpotent elements

Let $\mathcal{P}$ be a Hopf operad generated by primitive nilpotent elements, and let $\mathcal{Q}$ be the suboperad they generate, assumed to be Koszul. Write $\mathsf{Comm}$ for the commutative operad. Distributive-law conjecture. There exists a distributive law

$$\lambda:\mathcal{Q}\circ\mathsf{Comm}\to\mathsf{Comm}\circ\mathcal{Q}$$

that implies an isomorphism of symmetric collections

$$\mathcal{P}\simeq\mathsf{Comm}\circ\mathcal{Q}.$$

This predicts that the symmetric collection underlying such a Hopf operad is obtained by composing the commutative operad with the Koszul suboperad generated by its primitive nilpotent generators. The supplied text gives no information about whether the claim has been proved or disproved.