This question from last year (which in itself was anticipated on CSE two years prior) has a complete switchup of all non-pawn pieces on the board. The idea is to optimize the amount of moves needed to reach the given diagram.
In that puzzle, all of the pawns move to allow for fast switches. But what is the least possible pawn movement that allows for all pieces to legally be switched around? I found that only four pawns needed to be moved.
As such, a made a diagram that heavily restricts piece traffic flow, while only four pawns have moved in total. I consider my variation of the linked puzzle, hence the title of the question.
Thus, I ask this: From the starting position of a chess game, what is the fastest way to legally reach the following diagram? The board is from White's POV.