Compatibility conjecture for Poisson and Dunn–Lurie additivity

Let $n\geq 2$. Using the formality equivalences of Hopf operads $\mathbb{P}_n\cong\mathbb{E}_n$ and $\mathbb{P}_{n+1}\cong\mathbb{E}_{n+1}$, consider the diagram

\xymatrix{ \mathcal{A}\mathrm{lg}_{\mathbb{P}_{n+1}} \ar^-{\sim}[r] \ar^{\sim}[d] & \mathcal{A}\mathrm{lg}(\mathcal{A}\mathrm{lg}_{\mathbb{P}_n}) \ar^{\sim}[d] \\ \mathcal{A}\mathrm{lg}_{\mathbb{E}_{n+1}} \ar^-{\sim}[r] & \mathcal{A}\mathrm{lg}(\mathcal{A}\mathrm{lg}_{\mathbb{E}_n}) }

Compatibility conjecture. This diagram is commutative. The question is whether the Dunn–Lurie additivity functor is compatible with the Poisson additivity functor under the formality equivalences; the source does not establish the commutativity, so its resolution remains open.