I didn't spend much time but I think my first message already contained inaccuracies.
If the e-pawn were the promoted one, then there were two imaginable paths: via e6xf7xg8=N or via e4xf5xg6xh7-h8=N. However both seem impossible. The first option would save on a capture & move(s) but there is no way black could get a rook on f7 to be captured. The second option fails because that would assume that the rook from h8 is captured on f5 or g6 BEFORE the h7-pawn is captured, but then the rook could not get out.
Hence it was not the e-pawn that got promoted, but the f-pawn. It could go this way: f2-f4-f5xRg6xh7-h8=N. The black rook moves to g8 until the promoted knight left the promotion square, and then gets back out via h8 and towards f3 where it is captured by the e-pawn exRf3.
Assuming the abovementioned points were correct, a next step is to count the minimum number of moves needed to put the white pieces at their final position & possible paths
- King: 2 (e1-f2-g3)
- Queen: 3 i.e. 2 to reach a5 (d1-h5-a5 or d1-a4-a5) and 1 last move (Qa5-a6)
- Ra: 3 (Ra-e1 then up and one to the side, or Ra-a3 then 1 right and 1 up)
- Rh: 3 (similar as above)
- Bw: 1 (Bf1-b5)
- Bb: 3 (Bc1-b2-e5-h2)
- e-pawn: 1 (exRf3)
- f-pawn 5 (f2-f4-f5 xg6 xh7 - h8=N)
- promoted Knight 5 (for instance Nh8-f7-d6-c4-b6-a8 other paths possible too)
- other pawns: 4
Total: 30, leaving us only one spare move and lots of dependencies for the sequence
- move Nb6-a8 must be made before b7-b6 by black, and after the rook got out to be captured on f5 or g6; that rook could only get out via a5 (after a6XBb5)
- the black white-squared bishop could get out only after the promoted knight landed on a8, meaning also after a6xBb5, hence the bishop went via b7 to d6 to a2 (white has not the luxury to move the knight on b1 twice), and before the whire b- and c-pawns moved to their final position
I hope to have a bit more time in the coming days, and to find input/comments by others (I may be wrong in some of my assumptions)