The critical one-half conjecture for circle packings

Let $4\mathcal{S}4$ be a circle packing and let $s_0\in\mathcal{S}$. For $0<p<1$, let $E_{\mathcal{S},p}(s_0,r)$ denote the event that $s_0$ is connected to distance $r$ by open disks, and let $E_{\mathcal{S},p}(s_0,\infty)$ denote the event that it is connected to infinity. The critical one-half conjecture for circle packings.

$$\mathbb{P}(E_{\mathcal{S},1/2}(s_0,\infty))=0.$$

This would imply absence of infinite components at $p=1/2$ for several classes of planar graphs, including recurrent simple plane triangulations and suitable Benjamini–Schramm limits. It remains open even for triangulation circle packings with carrier $\mathbb{R}^2$ and uniformly bounded disk radii.