Puzzle 4 is the boxbox one. I'm still not seeing it. Maybe it's a sign that it's stumping me but it feels like there are at most two moves that aren't obviously wrong.
After you do the first move. The second move is a rotate that you have to do. It's very hard to see.
I got it now. With hindsight, I should've expected that to be an option since you can hold the piece in position for as long as you need to.
Edit: I went to play a normal game after replaying level 5 and the puzzle pieces were in my deck too.