BD additivity conjecture

Work over the complete topological $k\llbracket\hbar\rrbracket$-modules, with their symmetric monoidal structure, and let $\mathbbst{BD}_0$ and $\mathbbst{BD}_1$ denote the corresponding BeilinsonDrinfeld operads. Write $\EuScript{A}\mathsf{lg}_{\mathbbst{BD}_i}(\EuScript{M}\mathsf{od}_{\mathbbst{k}\llbracket\hbar\rrbracket}^\otimes)$ for the $\infty$-category of unital algebras over $\mathbbst{BD}_i$, and similarly for $\mathbbst{E}_1$-algebras. BD additivity conjecture. There is an equivalence of $\infty$-categories

$$\EuScript{A}\mathsf{lg}_{\mathbbst{BD}_1}(\EuScript{M}\mathsf{od}_{\mathbbst{k}\llbracket\hbar\rrbracket}^\otimes)\simeq \EuScript{A}\mathsf{lg}_{\mathbbst{E}_1}\big(\EuScript{A}\mathsf{lg}_{\mathbbst{BD}_0}(\EuScript{M}\mathsf{od}_{\mathbbst{k}\llbracket\hbar\rrbracket}^\otimes)\big).$$

This equivalence should recover Safronov's Poisson additivity equivalence after tensoring with $\mathbbst{k}$, equivalently after setting $\hbar=0$. It would imply that the stated strategy for quantizing ordinary Poisson structures always works, extending the known quantization of symplectic manifolds to this general categorical setting.