This answer is in progress. Also it has a gap, see OP's comment.
Currently we answered the question for all white pieces and the black dark-squared bishop.
The missed pieces balance. White: the queen and two pawns (maybe promoted). Black: a rook, the dark-squared bishop, and two knights.
White pawn b3 captured no pieces, because otherwise the white bishop could not reach a2. So white pawn c4 captured two pieces. None of them was the dark-squared bishop.
It is easy to see that all three white pieces were captured by pawns, via f7xg6 and a7xb6xc5. So black pawns c6 and d6 are originated from c7 and d7, respectively. Therefore the last move of white pawn d7 was a capture. That capture was e6xd7, because otherwise it was c6xd7, but then this pawn captured at least two pieces to reach vertical c, so White captured at least five pieces, a contradiction. Clearly, the captured piece was not the dark-squared bishop.
The white pawn from g2 could not reach vertical e or its left to be captured, because of the capture limit, so it was promoted. To reach the eight horizontal it captured a piece, because the vertical g is blocked by black pawn g6. The capture was at vertical f, because vertical h is blocked by black pawn h7. The captured piece was the dark-squared bishop, because it was not captured by three White's captures considered above.
Since we detected all four White's captures, white pawn d7 captured no more pieces, so it originates from e2. Therefore white pawn c4 originates from a2.
The white pawn from f2 could not reach vertical e or its left to be captured, because it captured no pieces, so it was promoted too. Both white pawn promotions were at f8, and so they were after the black pawn f7 freed vertical f by the capture f7xe6. So the captured piece was the white queen. Thus both promoted white pawns were captured by the black pawn from a7 via a7xb6xc5.
