Almost no percolation/almost full percolation dichotomy in the Bradonjić–Saniee model

Let $(a,p,\theta)$ be fixed with $a>1$, and let $G_{n,r}^2$ be the random geometric graph in the Bradonjić–Saniee bootstrap percolation model, with initially infected set $A_0$, final infected set $A_{\infty}$, and vertex set $\mathcal{P}$. Assume $\theta\neq \frac{1+p}{2}$. Almost no percolation/almost full percolation dichotomy. With high probability, either

$$\left|A_{\infty}\setminus A_0\right|=o(n)$$

or

$$\left|\mathcal{P}\setminus A_{\infty}\right|=o(n).$$

This conjecture asserts a dichotomy between an infection that spreads to only a negligible number of additional vertices and one that infects all but a negligible number of vertices. The paper proves it when $\theta>\frac{1+p}{2}$ or when $\theta<\theta_{\mathrm{local}}(a,p)$, but leaves the full range of parameters open.