Uniformity conjecture for measures with frame measures

Let $\mu$ be a finite Borel measure on $\mathbb{R}^d$. Say that $\mu$ is translationally absolutely continuous if, for every Borel set $F$ in the support of $\mu$ with $\mu(F)>0$ and every $a\in\mathbb{R}^d$, the translated restricted measure $T_a\mu|_{F+a}$ is absolutely continuous with respect to $\mu$. For $\alpha>0$, define

$$K(\alpha)=\left\{x\in\operatorname{supp}(\mu):\lim_{r\to0}\frac{\log\mu(B_r(x))}{\log r}=\alpha\right\}.$$

A measure has only one local dimension if all points in its support at which the local dimension exists have the same value. Uniformity conjecture. If $\mu$ has a frame measure, then $\mu$ must be translationally absolutely continuous and have only one local dimension. The conjecture expresses the expectation that a measure admitting a frame measure distributes its mass uniformly across its support. The source gives no examples with more than one local dimension and a frame measure, and no resolution is stated.