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 $p\to0$ it holds that $$\tau\le \exp\left(\frac{\pi^2}{6p}-\frac{c\log(1/p)}{\sqrt p}\right)$$ for some positive constant $c$, while the classical model is known to lack the logarithmic factor.