Cyclicity conjecture for the highest Hamiltonian of the quadratic Poisson structure

Let $h^{(N)}$ be the Hamiltonian of degree $N$ associated with the $N\times N$ Lax matrix for the Camassa–Holm peakon phase space, and let the quadratic Poisson structure be the bracket defined in the paper.

Cyclicity conjecture. The Hamiltonian $h^{(N)}$ is the cyclic variable of the quadratic Poisson structure.

Direct calculations establish this claim for $N=2,3,4$. The conjecture proposes that the same holds for general $N$, identifying the cyclic, Poisson-commuting variable within the polynomial algebra generated by the Hamiltonians.