  We study mixing times for the totally asymmetric simple exclusion process
(TASEP) on a segment of size $N$ with open boundaries. We focus on the maximal
current phase, and prove that the mixing time is of order $N^{3/2}$, up to
logarithmic corrections. In the triple point, where the TASEP with open
boundaries approaches the Uniform distribution on the state space, we show that
the mixing time is precisely of order $N^{3/2}$. This is conjectured to be the
correct order of the mixing time for a wide range of particle systems with
maximal current. Our arguments rely on a connection to last-passage
percolation, and recent results on moderate deviations of last-passage times.
