Chalker–Mehlig conjecture for the two-point eigenvector overlap function

Let $\mathcal{O}_N^{(2)}(z_1,z_2)$ denote the two-point function for eigenvector overlaps of an $N\times N$ complex Ginibre matrix. For points $z_1,z_2\in\mathbb{C}$, write $|z_1-z_2|$ for their separation.

Chalker–Mehlig conjecture. (i) For any two points $z_1,z_2$ such that $|z_1|<1$, $|z_2|<1$ and $|z_1-z_2|>0$,

$$\mathcal{O}_N^{(2)}(z_1,z_2)\, \stackrel{N\to\infty}{\longrightarrow}\, -\frac{1}{\pi^2}\cdot \frac{1-z_1\overline{z}_2}{|z_1-z_2|^4}.$$

(ii) For any $\omega\in\mathbb{C}$ and $z$ such that $|z|<1$,

$$N^{-2}\mathcal{O}_N^{(2)}\Bigl(z+\frac{1}{2}N^{-1/2}\omega,z-\frac{1}{2}N^{-1/2}\omega\Bigr)\sim -\pi^{-2}(1-|z|^2)\frac{1-(1+|\omega|^2)e^{-|\omega|^2}}{|\omega|^4},\qquad\text{as $N\to\infty$.}$$

These asymptotics concern the bulk two-point eigenvector-overlap statistics of the complex Ginibre ensemble. Chalker and Mehlig gave a mathematically motivated argument for them, but the source states that, to the authors' knowledge, the result had not yet been fully rigorously proved.