Hausdorff-dimensional packing conjecture for special-linear symmetries

Let $E\subseteq\mathbb{R}^2$ be a compact set with Hausdorff dimension

$$\dim_H(E)=\alpha.$$

Assume there is a set $R\subseteq\operatorname{SL}_2(\mathbb{R})$ of matrices such that $\theta(E)=E$ for every $\theta\in R$, and let $\dim_H$ denote Hausdorff dimension. Hausdorff-dimensional packing conjecture. If

$$\dim_H(R)>\frac{3\alpha}{2},$$

then $E$ is contained in a line. This predicts that a planar compact set with sufficiently many special-linear symmetries must be degenerate; the source motivates it by analogy with packing-set results over the reals, and no resolution is given.