Star-shaped connectedness of the root hybrid Mandelbrot set

Let $\mathbb{D}(2)$ denote the disc of radius $2$ centered at the origin, and let $(0,c_1)$ be a template pair whose associated critical orbit is infinitely well behaved. The parameter set in question is the set of all such pairs.

Star-shaped connectedness conjecture. The set of infinitely well-behaved pairs $(0,c_1)$ is star-shaped connected.

This conjecture is based on simulations following the proof that the set contains the disc centered at the origin of radius $1/4$; no resolution is given in the source.