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 $Per_1(1)$.
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