Local-limit conjecture for halfway permutation point processes

Let $(x,y)$ lie in the open unit disk. Define the rescaled point process

$$\Pi_n(x,y)=\left\{\frac{\sqrt n}{\sqrt{2\pi}(1-x^2-y^2)^{1/4}}\left(\sigma_G^n(i,0)-x,\sigma_G^n(i,1/2)-y\right):i\in{1,\mathellipsis,n}\right\}\subset\mathbb{R}^2.$$

Local-limit conjecture. There exists a rotationally symmetric, translation-invariant point process $Pi$ on $\mathbb{R}^2$ such that, for every $(x,y)$ in the open unit disk,

$$\Pi_n(x,y)\stackrel{d}{\to}\Pi.$$

This conjecture seeks the universal local point-process limit inside the Archimedean support of the halfway permutation; the paper presents it as an open problem.