Three-term progression distance conjecture for fractal measures

Let $u$ be a compactly supported probability measure on $\mathbb{R}^d$ satisfying

$$\nu(B(x,r))\leq C_Hr^\alpha$$

and

$$|\widehat{\nu}(\xi)|\leq C_F|\xi|^{-\beta/2}.$$

Assume that $\alpha>2d/3$. Three-term progression distance conjecture. Under appropriate quantitative assumptions on the parameters involved, the set

$$\left\{r>0:\exists x,u\ \oldtext{ with }\ |u|=r\ \oldtext{ and }\ x,x+u,x+2u\in\operatorname{supp}\nu\right\}$$

has positive Lebesgue measure. Earlier work established existence of three-term arithmetic progressions, and in some cases positive measure for their starting points; the conjecture asks for positive measure of the set of progression lengths in the stated fractal setting.