Reeb flows without simple global surfaces of section

We construct, for any given positive integer $n$, Reeb flows on contact integral homology 3-spheres which do not admit global surfaces of section with fewer than $n$ boundary components. We use a connected sum operation for open books to construct such a system. We prove that this property is stable with respect to $C^{4+\epsilon}$-small perturbations of the Hamiltonian given on the symplectization.