Symmetric local growth threshold conjecture for the Bradonjić–Saniee model

Let $(a,p,\theta)$ be fixed with $a>1$, and let $\theta_{\mathrm{local}}(a,p)$ denote the threshold for the appearance of large local, symmetrically distributed infectious outbreaks in the Bradonjić–Saniee bootstrap percolation model on $G_{n,r}^2$. Let $A_0$ be the initially infected set and $A_{\infty}$ the final infected set. Symmetric local growth conjecture. If

$$\theta>\theta_{\mathrm{local}}(a,p),$$

then with high probability almost no percolation occurs, namely

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

Together with the proved result that $\theta<\theta_{\mathrm{local}}(a,p)$ implies almost percolation, this conjecture identifies $\theta_{\mathrm{local}}$ as the threshold for almost percolation. It is presented as a strengthening of the almost-no/almost-full percolation dichotomy and remains open in the stated range.