The issue with my code was that I didn't make a push atomic, i.e. I would move boxes even if their ancestors couldn't be pushed. Making a list of candidate boxes to push solved this.
Here's the visualisation of the complete solution, though it doesn't show the last 100 frames or so. Please forgive me
Definitely a bit tedious, I had to "play" a whole session to spot bugs that I had. It took me far longer than average. I had buggy dissepearing boxes because of update order, I would reccomend a basic test case of pushing a line/pyramid of boxes in every direction.