Critical-radius ordering conjecture for transitive, Gaussian-analytic, and Poisson processes

Let $\Pi_\mathrm{L}$ be any transitive point process of intensity $1$, let $\Pi_\mathrm{GAF}$ be the zero set of a Gaussian analytic function with intensity $1$, and let $\Pi_\mathrm{P}$ be a Poisson point process with intensity $1$. For a point process $\Pi$ on $\mathbb{R}^2$, let $r_\mathrm{c}(\Pi)$ be the smallest radius for which the union of radius-$r$ balls centered at the points of $\Pi$ contains an infinite component. Critical-radius ordering conjecture.

$$r_\mathrm{c}(\Pi_\mathrm{L})<r_\mathrm{c}(\Pi_\mathrm{GAF})<r_\mathrm{c}(\Pi_\mathrm{P}).$$

This expresses the belief that more organized point processes have smaller critical radii; the source identifies it as an open problem comparing transitive, Gaussian-analytic, and Poisson models.