  We study the totally asymmetric simple exclusion process with open boundaries
in the high density and the low density phase. In the bulk of the two phases,
we show that the process on a segment of length $N$ exhibits cutoff at order
$N$, while in the intersection of the phases, the coexistence line, the mixing
time is of order $N^2$, and no cutoff occurs. In particular, we determine the
$\varepsilon$-mixing time in the coexistence line up to constant factors, which
do not dependent on $\varepsilon$. Combined with previous results on the
maximal current phase, this completes the picture on mixing times for the TASEP
with open boundaries.
