Aurell-Frisch-She conjecture for regular points of Burgers flow

Let $X=(X_x)_{x\in\mathbb R}$ be a self-similar process satisfying the growth condition needed to define the inviscid Burgers solution, and let $\mathcal L$ be the closure of the set of Lagrangian regular points at time $1$. Write $\operatorname{Dim}\mathcal L$ for its Hausdorff dimension. Aurell-Frisch-She conjecture. If $X$ is fractional Brownian motion with Hurst parameter $H\in(0,1)$, then

$$\operatorname{Dim}\mathcal L=H\quad\text{a.s.}$$

The paper explicitly says this conjecture remains open in general and discusses partial results.