Let $(W^{2n}, \Omega)$ be a symplectic cobordism from $(Y_+, \lambda_+, \omega_+)$ to $(Y_-, \lambda_-, \omega_-)$, where $(Y_+, \lambda_+, \omega_+)$ are $(Y_-, \lambda_-, \omega_-)$ are manifolds with stable Hamiltonian structures. Let $\gamma_+$ and $\gamma_-$ be Reeb orbits of $(Y_+, \lambda_+, \omega_+)$ and $(Y_-, \lambda_-, \omega_-)$ respectively. Let $\mathcal{M}^J(\gamma_+, \gamma_-, A)$ be the moduli space of punctured holomorphic curves in $W$ whose positive punctures are asymptotic to $\gamma_+$ and negative punctures are asymptotic to $\gamma_+$. Moreover, the holomorphic curves in $\mathcal{M}^J(\gamma_+, \gamma_-, A)$ have a fixed homology class $A$.
In the paper "Compactness results in Symplectic Field Theory", its authors define the moduli space of holomorphic buildings $\mathcal{M}^{J}_{br}(\gamma_+, \gamma_-, A)$. Moreover, they show that any sequence of curves in $\mathcal{M}^{J}(\gamma_+, \gamma_-, A)$ converges (after passing to a subsequence) to a limit in $\mathcal{M}^{J}_{br}(\gamma_+, \gamma_-, A)$. Let $\overline{\mathcal{M}^J(\gamma_+, \gamma_-, A)}$ be the compactification of $\mathcal{M}^{J}(\gamma_+, \gamma_-, A)$ by adding the limit points in $\mathcal{M}^{J}_{br}(\gamma_+, \gamma_-, A)$.
My question: Do we have $\overline{\mathcal{M}^J(\gamma_+, \gamma_-, A)}=\mathcal{M}^{J}_{br}(\gamma_+, \gamma_-, A)$? In other words, does any holomorphic building comes from the limit of a sequence of holomorphic curves? Is it possible that $\mathcal{M}^J(\gamma_+, \gamma_-, A)=\emptyset$ and $\mathcal{M}^{J}_{br}(\gamma_+, \gamma_-, A) \ne \emptyset$?
