The lower-threshold conjecture for standard bootstrap percolation

In the standard bootstrap percolation model, let $L\to\infty$ and $p\to 0$ simultaneously, and write $I(L,p)$ for the probability of percolation in an $L\times L$ box. Let

$$\lambda=\frac{\pi^2}{18}.$$

Lower-threshold conjecture. For any $\epsilon>0$, if

$$L<\exp\frac{\lambda-p^{1/2-\epsilon}}{p},$$

then

$$I(L,p)\to 0.$$

This conjecture refines the known sharp-threshold result by predicting a quantitative correction to the lower metastability threshold. It is motivated by the relationship between global percolation and indefinite growth in the local model; the supplied source does not state a resolution.