Quasi-isomorphism for different Boardman–Vogt models

Establish that any homotopy equivalence between different choices of Boardman–Vogt models from an A∞ operad to the operad of chains on stable disc moduli spaces induces quasi-isomorphic A∞-categories Fuk^f_{K,ρ} for every compact subset K and potential value ρ.

Background

The paper constructs the A∞-category Fuk^f_{K,ρ} by choosing a Boardman–Vogt model f: A → O, where O is the operad of chains on stable moduli spaces of discs. The objects and morphism spaces are shown to be independent of this choice, but the authors do not establish invariance of the resulting A∞-category up to quasi-isomorphism.

The unresolved problem is to prove that homotopy-equivalent choices of Boardman–Vogt model yield quasi-isomorphic A∞-categories. Such a result would show that the A∞-category with support is canonical up to the appropriate homotopical equivalence, rather than depending on the auxiliary operadic model.