I found that the book "Amusements In Chess," by Henry Ernest Dudney, has been put onto the web in the form of a website. In the section "Various Chess Puzzles, I came across this particular puzzle that intrigues me.
The author gives this legal position and asks for a way to reach it. The White king is on c7, and the Black king is on g1.
On the linked solution page near below it, it is said that: "The order of the moves is immaterial, and this order may be greatly varied. But, although many attempts have been made, nobody has succeeded in reducing the number of my moves."
The author provides his solution in 43 moves, which I have transcribed from descriptive notation, replayable here.
- f4 c6 2. Kf2 Qa5 3. Ke3 Kd8 4. f5 Kc7 5. Qe1 Kb6 6. Qg3 Na6 7. Qb8 h5 8. Nf3 Rh6 9. Ne5 Rg6 10. Qxc8 Rg3+ 11. hxg3 Kb5 12. Rh4 f6 13. Rd4 fxe5 14. b4 exd4+ 15. Kf4 h4 16. Qe8 h3 17. Nc3+ dxc3 18. Ba3 h2 19. Rb1 h1=Q 20. Rb2 cxb2 21. Kg5 Qg1 22. Qh5 Ka4 23. b5 Rc8 24. b6 Rc7 25. bxc7 b1=B 26. c8=R Qc7 27. Bd6 Nb4 28. Kg6 Ka3 29. Ra8 Kb2 30. a4 Qgb6 31. a5 Kc1 32. axb6 Kd1 33. bxc7 Ke1 34. Kf7 Nh6+ 35. Ke8 Ba2 36. f6 Bg8 37. f7 Kxf1 38. c8=B Nd5 39. Bb8 Nc7+ 40. Kd8 Ne8 41. fxe8=R Nf7+ 42. Kc7 Nd8 43. Qf7+ Kg1
However, is this truly the best possible? I do not know. So, the question here should be obvious. Can lower than 43 moves, or 86 ply, be achieved, or can it be proved as optimal?