The one-third regularity threshold for Euler solutions with anomalous diffusion

Let $d\in\{2,3\}$. Let $u\in C^0([0,1];C^\alpha(\mathbb T^d))\cap C^\alpha([0,1];C^0(\mathbb T^d))$ be a weak solution of the incompressible Euler equations, for some Hölder exponent $\alpha\in(0,1)$. Assume that, for every initial condition $\theta_{\mathsf{in}}\in\dot H^1(\mathbb T^d)$, the family of solutions $\{\theta^\kappa\}_{\kappa>0}$ of the passive scalar equation with velocity field $u$ displays anomalous diffusion continuously in time: the anomalous-diffusion condition holds and the time dissipation measure $\mathcal E(dt)$ is non-atomic. One-third regularity threshold conjecture. Then

$$\alpha\leq\frac13.$$

The conjecture asserts that the Onsager-critical exponent is a rigidity threshold for Lagrangian turbulent diffusion: anomalous diffusion for every initial condition should be incompatible with Euler velocity fields having Hölder regularity above $1/3$. Its resolution would connect anomalous scalar dissipation with Onsager super-criticality, while the stated implication remains open.