The smoothness conjecture for supercritically dissipative Hilbert models

Let $\theta_0$ be initial data satisfying the hypotheses of the Hölder $1/2$ a priori estimate conjecture, and consider the supercritically dissipative Hilbert model

$$\partial_t\theta+H\theta\,\partial_x\theta+\Lambda^\gamma\theta=0,$$

with $\gamma>1/2$, where $\Lambda=(-\Delta)^{1/2}$. A smoothness conjecture. The vanishing-viscosity limits lie in

$$L^\infty([0,\infty);C^\infty).$$

This is presented as a consequence of the conjectured Hölder estimate and the regularity theory for the dissipative equation; it remains open in the source.