The lattice bacteria puzzle

We're going to play a game with some bacteria sitting on a grid. At any point, you can select one of them, and if the spaces, one above and one to the right, are both empty, you can make it replicate, populating both of those spots with its children and vacating the spot where it came from. The rule is that only one cell is allowed at each point. So, if one of those two target spots is currently blocked for a given bacterium, it can't replicate. But, as soon as both of them clear out, it's free to do so. Here's my puzzle for you. Suppose you begin with just one cell at the origin and your goal is to eventually clear out this box here, the one with corners at 0 03 33 and 3 0. What is the smallest number of moves required to do so? To be clear, all 16 of these lattice points have to end up empty. This is part of a monthly series of puzzles that I'm doing in collaboration with Moath. The mathematician Peter Winkler will host a session there to explain the solution. And I'll post my own video of the solution here sometime next month.