Sensitive bootstrap percolation second term

In modified two-neighbour bootstrap percolation in two dimensions each site of ${\mathbb{Z}}^2$ is initially independently infected with probability $p$ and on each discrete time step one additionally infects sites with at least two non-opposite infected neighbours. In this note we establish that for this model the second term in the asymptotics of the infection time $\tau$ unexpectedly scales differently from the classical two-neighbour model, in which arbitrary two infected neighbours are required. More precisely, we show that for modified bootstrap percolation with high probability as $p\to 0$ it holds that $$\tau \le \exp (\frac{{\pi }^2}{6p}-\frac{c\log (1\mathrm{/}p)}{\sqrt{p}})$$ for some positive constant $c$, while the classical model is known to lack the logarithmic factor.