Connected excision conjecture for sutured monopole Floer contact elements

Let $(M_1,\gamma_1)$ and $(M_2,\gamma_2)$ be balanced sutured manifolds equipped with contact structures as in Theorem 1, and let $(M,\gamma)$ be a connected balanced sutured manifold with a diffeomorphism

$$g:\partial M\xrightarrow{\cong}\partial M_1\sqcup\partial M_2$$

that sends $\gamma$ to $\gamma_1\cupgamma_2$. The corresponding closures are related by a Floer excision cobordism, inducing a map analogous to $F$.

Connected excision conjecture. Theorem 1 continues to hold when the disjoint union $(M_1\sqcup M_2,\gamma_1\cupgamma_2)$ is replaced by the connected balanced sutured manifold $(M,\gamma)$ described above; in particular, the excision map preserves the associated contact elements up to multiplication by a unit.

The conjecture addresses the connected case not covered by the theorem proved in the paper. Some evidence is suggested by slicing and torus $1$-handle operations, which give the relevant cobordism a weak symplectic structure, but the asserted contact-element preservation remains conjectural.