Is there any reference in the literature about results regarding SFT compactness for a neck-stretch in the context of Floer homology of a symplectomorphism $\phi \colon (M,\omega) \to (M,\omega)$, where $M$ is a closed symplectic manifold?

I would be interested in stretching $M$ along a tubular neighbourhood of a contact type codimension $1$ closed submanifold $S$.

I can see two problems:

if we consider the classical interpretation of $HF^*(\phi)$ as in [DS94] then we are considering $J$-holomorphic strips $u \colon \Bbb R \times [0,1] \to M$ with boundary condition $u(s,1)=\phi u(s,0)$. The target space would be closed, the submanifold $S$ would be closed but the domain wound be biholomorphic to a closed disk with two punctures on the boundary. Since we can assume that $$\text{lim}_{s \to \pm \infty} u(s,t) = x_{\pm} \in \text{Fix}(\phi)$$ then $u$ would extend to a map from the disk to $M$. But the only version of SFT for curves with boundary I’m aware relies on some control on where these boundaries are mapped (some lagrangian boundary conditions say)

Therefore I focused on the interpretation of $HF^*(\phi)$ as the count of horizontal sections of the symplectic mapping torus $\Bbb R \times M_{\phi}\to \Bbb R \times S^1$ (See the PhD thesis of Seidel for example). In this scenario, we are interested in $J$-hol sections of the symplectic mapping torus (call it for brevity $T_{\phi}$) whose domain is $\Bbb R \times S^1$. The upside now is that the domain doesn’t have boundary but the drawback is that $T_{\phi}$ is not compact and (assuming that $\phi \colon S\to S$) the neighbourhood on which I want to stretch would be a tubular neighbourhood of $\Bbb R \times S_{\phi}$ which again, is not compact.

On [CM05] the authors state the compactness theorem for curves into a closed symplectic manifold stretched along a closed stable hypersurface $M$, but then claims that the proof goes through in the case $X$ is a symplectic cobordism $X$ (which is not compact since we attach suitable ends to its boundary components) and $M$ is a closed stable hypersurface.

Can someone point me out where compactness of $M$ is used in that paper?

My hopes are that since we are dealing with finite energy sections maybe we can circumvent the fact that $\Bbb R \times S_{\phi}$ is not compact and rely instead on the closeness of $S_{\phi}$

And more importantly,

Are there instances of the result I’m interested in, already available in the literature?

REFERENCES

[CM05] Cieliebak, K.; Mohnke, K. Compactness for punctured holomorphic curves. Conference on Symplectic Topology. J. Symplectic Geom. 3 (2005), no. 4, 589--654. MR2235856

[DS94] Dostoglou S., Salamon D. Self-dual instantons and holomorphic curves. Annals of Mathematics 139.3 (1994): 581-640.