Mating quadratic maps with the modular group II

Abstract: In 1994 S. Bullett and C. Penrose introduced the one complex parameter family of $(2:2)$ holomorphic correspondences $\mathcal{F}_a$: $$\left(\frac{aw-1}{w-1}\right)^2+\left(\frac{aw-1}{w-1}\right)\left(\frac{az+1}{z+1}\right) +\left(\frac{az+1}{z+1}\right)^2=3$$ and proved that for every value of $a \in [4,7] \subset \mathbb{R}$ the correspondence $\mathcal{F}_a$ is a mating between a quadratic polynomial $Q_c(z)=z^2+c,\,\,c \in \mathbb{R}$ and the modular group $\Gamma=PSL(2,\mathbb{Z})$. They conjectured that this is the case for every member of the family $\mathcal{F}_a$ which has $a$ in the connectedness locus. We prove here that every member of the family $\mathcal{F}_a$ which has $a$ in the connectedness locus is a mating between the modular group and an element of the parabolic quadratic family $Per_1(1)$.