1 a4 a5 2 b4 b5 3 axb5 axb4 4 c3 c6 5 bxc6 bxc3 6 cxd7+ Bxd7 7 h4 cxd2+ 8 Bxd2 h5 9 g4 g5 10 hxg5 hxg4 11 f3 f6 12 gxf6 gxf3 13 fxe7 Bxe7 14 Bc1 fxe2 15 Bxe2 Bc8 16 Bf1 Bf8
Sorry, this is with a rotated board...
I found a solution in 39 moves, but it probably can shorter
1 a4 a5 2 b4 b5 3 axb5 axb4 4 c3 c6 5 bxc6 bxc3 6 cxd7+ Nxd7 7 h4 cxd2+ 8 Nxd2 h5 9 g4 g5 10 hxg5 hxg4 11 f3 f6 12 gxf6 gxf3 13 fxe7 Nxe7 14 Kf2 fxe2 15 Nxe2 Kf7 16 Ke3 Kf6 17 Kd4 Kf5 18 Kc4 Kg4 19 Kb5 Ra2 20 Ne4 Kf3 21 Nf6 Kf2 22 Ng8 Nd5 23 Kc6 Nc3 24 Nd4 Qh4 25 Kc7 Ne5 26 Qh5 Ke1 27 Qe8 Kd1 28 Nc6 Qe1 29 Kd8 Nf3 30 Nb8 Ng1 31 Rh6 Rb2 32 Ra8 Rb1 33 Ba6 Ra1 34 Rg6 Rh1 35 Rh6 Bh3 36 Bc8 Nb1 37 Rh8 Bh6 38 Ba3 Bc1 39 Bf8 Bf1
Shortened to 37 moves:
1. a4 a5 2. b4 b5 3. axb5 axb4 4. c3 c6
5. bxc6 bxc3 6. cxd7+ Nxd7 7. h4 cxd2+ 8. Nxd2 h5
9. g4 g5 10. hxg5 hxg4 11. f3 f6 12. gxf6 gxf3
13. fxe7 fxe2 14. Nxe2 Nxe7 15. Rg1 Rh1 16. Rg8 Rb8
17. Rh8 Rb1 18. Ra8 Ra1 19. Kf2 Kf7 20. Ke3 Kg6
21. Kd4 Kg5 22. Qa4 Ne5+ 23. Kc5 Qa5+ 24. Kd6 Bh6
25. Nd4 Kf4 26. Ba3 Ke3 27. Bh3 Ba6 28. Ne4 Nf3
29. Kd7 Qe1 30. Nc6 Ke2 31. Kd8 Bc1 32. Bc8 Nd5
33. Nb8 Ng1 34. Qe8 Kd1 35. Nf6 Nc3 36. Ng8 Nb1
37. Bf8 Bf1