Small-ball asymptotic conjecture for line visibility in the hyperbolic plane

Fix a point $o\in\mathbb{H}^2$. For each $r>0$, let $p_r$ be the least $p\in[0,1]$ such that every random closed set $\mathcal Z\subset\mathbb{H}^2$ with isometry-invariant law and

$$\mathbf{P}[B(o,r)\subset\mathcal Z]>p$$

has positive probability of containing a hyperbolic line. Theorem~ implies that $p_r<1$ for every $r>0$. Small-ball asymptotic conjecture.

$$\limsup_{r\searrow 0}\frac{1-p_r}{r}<\infty.$$

This asks for a linear upper bound on the deficit $1-p_r$ as the ball radius tends to zero, quantifying the high-probability condition needed for line visibility.