The rigid-transformation packing conjecture

Let $E\subset \mathbb{R}^d$ and let $\Theta\subset E(d)$ be Borel sets, where $E(d)$ denotes the group of rigid transformations of $\mathbb{R}^d$. Write $\dim_{\mathcal{SF}}E$ for the spherical Furstenberg dimension of $E$. If

$$\dim_{\mathcal{SF}}E+\dim_{\mathcal{H}}\Theta>\frac{d^2+d}{2},$$

then

$$\mathcal{L}^d(\Theta(E))>0.$$

Rigid-transformation packing conjecture. Under this dimension hypothesis, the set obtained by applying the transformations in $\Theta$ to $E$ has positive $d$-dimensional Lebesgue measure. This conjecture is proposed as a plausible consequence of the preceding theorem; no resolution is given in the source.