Naturality conjecture for filtered Seiberg–Witten cobordism maps and ECH

Fix $L>0$. Let $(X_i,\lambda_i)$ be exact symplectic cobordisms from $(Y^i_-,\lambda^i_-)$ to $(Y^i_+,\lambda^i_+)$, with cobordism-admissible almost complex structures $J_i$, for $i=0,1$. Suppose there are a 4-dimensional manifold with corners $W$, a $1$-form $\lambda_W$, and disjoint embedded $3$-manifolds with corners $Z_\pm\subset\partial W$, together with embeddings $f_i:W\hookrightarrow X_i$ satisfying the stated compatibility of the contact forms, containment of all Reeb orbits of action less than $L$ in $f_i(Z_\pm)$, and containment of all relevant broken $J_i$-holomorphic curves in $\mathbb{R}\times\operatorname{int}(f_i(W))$.

Naturality conjecture. The diagram of filtered Seiberg–Witten Floer cohomology groups, with horizontal maps given by the chain maps of and vertical maps given by the canonical identifications of monopoles with ECH generators of action at most $L$ inside $Z_\pm$, is commutative.

This conjecture would show that the cobordism maps used to define the ECH isomorphisms are independent of the embedding data, and hence that the resulting direct-limit isomorphism is canonical. Although the conjecture is open, the paper proves enough cases in its appendix to remove the potential dependence on the embedding data.