Janicki and Woyczynski's Hausdorff-dimension conjecture for Lagrangian regular points

Let $Z$ be the Lévy stable process and let $4\mathcal{L}_t4$ denote the set of Lagrangian regular points at time $t$, as defined through the Hopf–Cole solution of the inviscid Burgers equation. The stability parameter satisfies $\alpha\in(1,2]$.

Janicki and Woyczynski's conjecture. For every $t>0$, one has

$$\operatorname{Dim}_H\mathcal{L}_t=1/\alpha\quad\text{a.s.}$$

The conjecture concerns the fractal dimension of particles that have not been shocked by time $t$. The source does not state a resolution; however, it immediately discusses a result showing that the conjecture is false in general when there are positive jumps.