Prove compatibility of modular-operadic Massey products with FA-module structures

Establish whether the Massey products in the modular-operad setting can be chosen compatibly with the FA-module structure on the relevant Lie graph-homology constructions, thereby determining whether any two statements in Theorem 1.3 imply the third.

Background

Theorem 1.3 presents three relationships among filtered pieces of the Feynman transform of Lie graph homology and compactly supported configuration-space cohomology. The authors explain that these relationships suggest an underlying compatibility between modular-operadic Massey products and FA-module structures.

Such compatibility would provide a conceptual implication among the three statements, rather than requiring the newly proved statements to be established independently. The paper does not prove that the required compatible choice of Massey products exists.

References

Our results, particularly Theorem 1.3, suggest such Massey products can be chosen to be compatible with the FA-module structure. If this were known to be true, it would be possible to conclude that any two of the statements in Theorem 1.3 imply the third.

— From configuration spaces to graph complexes via FA-modules  (2609.04033 - Carter et al., 3 Sep 2026) in Section 1.4, “Connections and Future Directions,” p. 4