Chain-morphism and commutativity of the tensor-product isomorphism g in the almost Künneth formula

Prove that the filtered Λ-module isomorphism g: CF_*(L0′ ∪ L1′, Hε″) ⊗Λ CF_*(L2′, Hε′) → CF_*(L0 ∪ L1 ∪ L2, Hε), constructed in Proposition 4.9 (the almost Künneth formula for link Floer homology), is a chain map and that it commutes with the continuation maps induced by the Hamiltonian isotopies considered in the paper.

Background

To relate spectral invariants across different link configurations, the authors construct a filtered Λ-module isomorphism g that mirrors an almost Künneth-type behavior for link Floer complexes. While they establish the filtered module isomorphism, they cannot prove it respects the differentials or interacts functorially with continuation maps.

They explicitly conjecture that g is a chain morphism and commutes with continuation maps, which would strengthen the ‘almost Künneth’ framework and streamline the analysis of spectral invariants and their continuation behavior.

References

In contrast to Proposition \ref{prop.commutativity with cont maps}, we did not manage to prove that the above isomorphism is a chain morphism or that it commutes with the continuation maps. We conjecture that this is true, but we will not need it for our purposes.

— On link quasimorphisms on the sphere and the equator conjecture  (2509.14996 - Serraille et al., 18 Sep 2025) in Section 4.3 (An almost Künneth formula for link Floer homology)