The one-sided fractional Laplacian bound conjecture for vanishing-viscosity solutions

Let $\theta_0$ satisfy the hypotheses of the Hölder $1/2$ a priori estimate conjecture, and let $\theta$ be a vanishing-viscosity limit of solutions of

$$\partial_t\theta+H\theta\,\partial_x\theta=0.$$

Here $\Lambda=(-\Delta)^{1/2}$ is the one-dimensional fractional Laplacian. A one-sided fractional Laplacian bound conjecture. There is a nonnegative function $A(t)=A(t,\theta_0)$ such that, for every $t>0$,

$$\Lambda\theta(t,x)\geq-A(t).$$

The conjecture has the geometric interpretation that singular cusps forming in finite time open only downward, and is discussed as closely connected with the Hölder regularization conjecture. It is open in the source.