The dg–$A_\infty$ module-category equivalence conjecture

Let $S$ be the underlying scheme, and let $\mathcal{B}$ be a sheaf of $\mathbb{Z}$-graded differential-graded $\mathcal{O}_S$-algebras and $\mathcal{A}$ a sheaf of $\mathbb{Z}$-graded $A_\infty$ $\mathcal{O}_S$-algebras. Write $\operatorname{D}_{\operatorname{pe}}(\operatorname{Mod}_{\infty,\mathbb{Z}}\mathcal{A})$ and $\operatorname{D}_{\operatorname{pe}}(\operatorname{Mod}_{\mathbb{Z}}\mathcal{B})$ for the perfect derived categories of strictly unital graded $A_\infty$-modules and graded dg-modules, respectively. If $\mathcal{B}$ is graded-quasi-isomorphic to $\mathcal{A}$, then the dg–$A_\infty$ module-category equivalence conjecture. There is a $\mathbb{P}(S)$-linear equivalence

$$\operatorname{D}_{\operatorname{pe}}(\operatorname{Mod}_{\infty,\mathbb{Z}}\mathcal{A})\cong \operatorname{D}_{\operatorname{pe}}(\operatorname{Mod}_{\mathbb{Z}}\mathcal{B}).$$

Such an equivalence would identify the perfect module categories associated with quasi-isomorphic graded dg and $A_\infty$ algebra sheaves; the paper states that the conjecture is believed to be true, but does not prove it.