Star-shapedness conjecture for the Hausdorff-dimension-greater-than-one locus

Let $\mathcal{M}$ be the Mandelbrot set, and write

$$\left(\mathbb{C} \setminus \mathcal{M}\right)_{>1}=\{c\notin\mathcal{M}\mid \operatorname{dim}_{\mathrm{H}}J_c>1\},$$

where $J_c$ is the Julia set of the quadratic map with parameter $c$. Star-shapedness conjecture. The set $\left(\mathbb{C} \setminus \mathcal{M}\right)_{>1}$ is star-shaped centered at $0$. This is suggested by numerical evidence concerning the Hausdorff-dimension-one locus outside the Mandelbrot set; the source does not establish the claim.