"This is false. Given a low entropy state, the past is either a lower entropy state or a same-entropy state, and there's no time-reversal operator for thermodynamic processes (except those carried out at constant entropy) regardless of boundary conditions."
That's a common misunderstanding evident in a number of the answers and comments above, that I think is worth addressing at greater length than is possible in a comment. (My apologies if this is against the rules on here.)
There are two aspects to the laws of physics within a region of spacetime that need to be considered: the kinematic laws, and the statistical laws.
The kinematic laws are those mentioned by the OP - that for any given forwards-in-time history, you can reverse all the final velocities of the particles, call them initial velocities, and trace the same history backwards. That's true, and agreed by all sides - at least in the classical physics version.
The entire trajectory of every particle throughout the period is fully determined by their joint position and velocity at any given instant, which includes both the start of the period and its end, and reversing the velocities of all particles at any instant reverses the trajectories. The trajectories in the bulk of the spacetime region are fully determined by the trajectories at the boundary.
The statistical laws are about the number of possible trajectories fulfilling particular macroscopic conditions. The idea is that the number of trajectories exhibiting 'normal' behaviour so vastly exceeds the number where the second law is violated that it becomes a virtual certainty that things will proceed as expected. While violations are possible, they are exceedingly improbable. Thus, people try to derive the second law as a statistical effect. It isn't.
Let's consider a classic example - a large number of gas molecules starting in one half of the chamber. We specify the positions of the particles on the past boundary, but we haven't said anything about their velocities. So we suppose they are selected uniformly from all the possibilities.
Now for each starting combination of positions and velocities, the entire subsequent history is determined. But over the range of all possible choices of velocities, there are a vast, vast number of possible trajectories.
In some, all the particles finish in the same half of the box. In some, they all end up in the same hundredth of the box, crammed into one tiny corner. But the number of initial states where they're spread out between the two halves vastly exceeds the number where they're in the same half, which even more vastly exceeds the number where they're in the same one hundredth. So given a uniform choice over our range of possible starting states - specified positions, arbitrary velocities - it is virtually certain that we've got one of the spread-out ones rather than a huddle-together one. This is the statistical argument's explanation for the second law.
However, this argument only works if you apply it to the initial states - when the full trajectories are determined by their values at any time. Thus, we can equally easily assert that at the end of the experiment all the gas molecules are in the same half of the box, and ask how they got there. The number of time-reversed trajectories satisfying the time-reversed constraints is exactly the same number. So there are vastly more choices of final velocities that are preceded by molecules spread out fairly equally between the two halves than there are choices where the molecules started in the same half, or an even smaller region.
So if we take the statistical argument seriously, then we ought to expect that setting a low-entropy condition on the final state would be preceded by an entropy decrease! This is what the statistical argument tells us. So the statistical argument contradicts the second law.
Note, I am not saying that the second law is wrong. I am saying that the statistical argument does not imply or explain it.
The statistical argument simply counts trajectories - but the number of time-reversed trajectories is identical to the number of forwards-time trajectories, so the statistical argument is as time-reversal symmetric as the kinematic argument is. We have to look elsewhere for an explanation.
The second law says that in the bulk of each spacetime region the entropy does not decrease. This implies that the entropy at the end time is always equal to or greater than the entropy at the start time. Any statement about the bulk is also a statement about the boundary, and wording it this way directs us towards an understanding.
High entropy requires no explanation. Statistically, virtually all trajectories are high-entropy. The big question we really need to answer is where does the low entropy come from? On statistical grounds, the starting conditions of our thermodynamic experiments are fantastically unlikely. Statistical arguments cannot explain them. But they're clearly observed, so we need an explanation.
Not only are they observed to happen, we also observe that they always happen on the past boundary, never the future one. If we constrain the past boundary to a low-entropy state, leave the future boundary free, statistical arguments predict exactly the sort of events that we commonly see. But if we constrain the future boundary to a low-entropy state and leave the past free, statistical arguments make the wrong prediction. Instead of predicting an even lower entropy initial state, they predict entropy decrease.
The second law states that the lowest entropy is always on the past boundary of any experiment. This cannot be explained by anything going on inside our region. It isn't explained by either the kinematic or statistical rules that apply inside the region, and they fully determine everything that happens inside the bulk. So it has to be something outside the region. Something happened in the deep past to start the universe off in an extremely low entropy state, and every instance of low entropy we ever observe experimentally originates there.
If we take the position that low entropy always originates in the past, then seeing low entropy on the future boundary, we can legitimately conclude that the only place it could have come from to get there is the past boundary, through the bulk of our experiment, and thus predict the initial conditions to be of even lower entropy. Statistically, that's fantastically unlikely. But the start of the universe is fantastically unlikely, so that's not a problem.