All-orders Hamiltonian conjecture for rotational water waves

Let $\epsilon$ and $\delta$ be the perturbation parameters, and let $\mathcal{H}_{nm}$ denote the Hamiltonian contribution of bidegree $(n,m)$ in the grading with respect to $(\epsilon,\delta)$. All-orders Hamiltonian conjecture. The full system is Hamiltonian with respect to

$$\mathcal{H}=\bigoplus_{n=0}^{\infty}\bigoplus_{m=0}^{\infty}\mathcal{H}_{nm}.$$

The preceding perturbative calculations exhibit Hamiltonian formulations through the displayed orders and suggest an infinite hierarchy of Hamiltonians, but the all-orders assertion is not established in the supplied text.