Equidistribution for matings of quadratic maps with the Modular group

We study the asymptotic behavior of the family of holomorphic correspondences $\lbrace\mathcal{F}_a\rbrace_{a\in\mathcal{K}}$, given by $$\left(\frac{az+1}{z+1}\right)^2+\left(\frac{az+1}{z+1}\right)\left(\frac{aw-1}{w-1}\right)+\left(\frac{aw-1}{w-1}\right)^2=3.$$ It was proven by Bullet and Lomonaco that $\mathcal{F}_a$ is a mating between the modular group $\operatorname{PSL}_2(\mathbb{Z})$ and a quadratic rational map. We show for every $a\in\mathcal{K}$, the iterated images and preimages under $\mathcal{F}_a$ of nonexceptional points equidistribute, in spite of the fact that $\mathcal{F}_a$ is weakly-modular in the sense of Dinh, Kaufmann and Wu but it is not modular. Furthermore, we prove that periodic points equidistribute as well.