Rough local smoothing conjecture for Ahlfors–David regular measures

Suppose that $\mu$ is an Ahlfors--David regular probability measure supported on $[1,2]$ and satisfying

$$\mu(B(t,r))\sim r^\beta$$

for $r<\operatorname{diam}(\operatorname{supp}\mu)$. Rough local smoothing conjecture. Then

\left\\|\left(e^{\pm it\sqrt{-\Delta}}\phi_j\left(\sqrt{-\Delta}\right)f\right)(x)\right\\|_{L^{q_{\mathrm{LS},\beta}}_x(\mathbb{R}^d;L^{q_{\mathrm{LS},\beta}}_t(d\mu))}\lessapprox \\|f\\|_{L^{q_{\mathrm{LS},\beta}}}.

This is presented as a more general measure-theoretic formulation of the rough local smoothing conjecture, and the paper states that the set formulation follows from it for Ahlfors--David regular sets; its status is open.