The Mandelbrot homeomorphism conjecture for modular matings

Let $\mathcal{C}_{\Gamma}$ be the connectedness locus of the family $\mathcal{F}_a$, and set

$$M_{\Gamma}=\mathcal{C}_{\Gamma}\cap\{z:|z-4|\leq 3\}.$$

Mandelbrot homeomorphism conjecture. The Mandelbrot set $M$ is homeomorphic to $M_{\Gamma}$. The family $\mathcal{F}_a$ has been shown to mate parabolic quadratic maps with $\operatorname{PSL}(2,\mathbb{Z})$, and the corresponding parabolic Mandelbrot set is known to be homeomorphic to $M$; proving the asserted homeomorphism for $M_{\Gamma}$ remains the stated goal.