Full percolation threshold conjecture for the Bradonjić–Saniee model

Let $(a,p,\theta)$ be fixed with $a>1$, and let $T_n^2$ be the two-dimensional torus. In the Bradonjić–Saniee bootstrap percolation model on the random geometric graph $G_{n,r}^2$, let $\theta_{\mathrm{islands}}(a,p)$ be the threshold for the disappearance of the relevant symmetric islands of uninfected vertices and let $\theta_{\mathrm{local}}(a,p)$ be the threshold for large local, symmetrically distributed infectious outbreaks. Write $\mathcal{P}$ for the vertex set and $A_{\infty}$ for the final infected set. Full percolation conjecture. If

$$\theta<\min\left(\theta_{\mathrm{islands}}(a,p),\theta_{\mathrm{local}}(a,p)\right),$$

then with high probability full percolation occurs:

$$\mathcal{P}=A_{\infty}.$$

The conjecture identifies the disappearance of symmetric uninfected islands and the occurrence of local growth as the two conditions expected to guarantee infection of every vertex. The paper proves that $\theta>\theta_{\mathrm{islands}}(a,p)$ leaves some vertices uninfected and proves almost percolation below $\theta_{\mathrm{local}}(a,p)$, but the full-percolation claim remains open.