Dichotomy conjecture for Hamiltonian conservation in Euler-$\alpha$

Consider the Euler-$\alpha$ equations on $\mathbb{T}^2$ or $\mathbb{T}^3$, and let $u$ be a mean-zero weak solution in $C_t^0H_x^1\cap L_t^3B_{3,\infty}^s$. Its Hamiltonian is

$$\mathcal{H}_{\alpha}(t)=\|u(t)\|_{L^2}^2+\alpha^2\|\nabla u(t)\|_{L^2}^2.$$

Euler-$\alpha$ Hamiltonian dichotomy conjecture. Weak solutions obey the following dichotomy: if $s>1$, every weak solution in $C_t^0H_x^1\cap L_t^3B_{3,\infty}^s$ conserves $\mathcal{H}_{\alpha}$; if $s<1$, there exist weak solutions in $C_t^0H_x^1\cap L_t^3B_{3,\infty}^s$ that do not conserve $\mathcal{H}_{\alpha}$.

The first part is supported by the paper's energy-conservation lemma, while the second part is the conjectural flexibility statement. Together they propose a sharp threshold at $s=1$ for Hamiltonian conservation in this formulation of Euler-$\alpha$.