  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.
