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It

is possible.

Proof

game: 1. Nf3 Nf6 2. Nh4 Nd5 3. Ng6 Nc6 4. Nxh8 Nd4 5. Ng6 Nb3 6. Ne5 Nf4 7. g3 Nxc1 8. Bg2 Ng6 9. Be4 Nh8 10. Rg1 Nb3 11. a3 let the queen out while preserving as many stalling moves as possible Nd4 12. Nc3 Ne6 13. Qb1 Ng5 14. Qa2 Ne6 15. Qd5 Rb8 16. Qc6 bxc6 17. Nd5 Ba6 18. Bf3 Bd3 19. Nc4 Bf5 20. Ne5 Bh3 21. Be4 Bf1 it must be the bishop, a knight would be too slow to reach their final position from here 22. Rg2 Qc8 23. Nf3 Rb6 24. Rg1 cxd5 25. Rg2 dxe4 26. Rg1 Qa6 27. Rg2 Kd8 28. Rg1 exf3 29. Rg2 Kc8 30. Rg1 Kb7 31. Rg2 Kc6 32. Rg1 Kd5 33. Rg2 Ke4 34. Rg1 Kf5 35. Rg2 Kg4 36. Rg1 Kh3 37. Rg2 Rb3 38. Rg1 Nd8 39. Rg2 fxg2 from here black needs to finish before white's remaining moves are used up 40. a4 g1=R 41. a5 Kg2 42. cxb3 Rh1 43. b4 Kg1 44. b5 Bg2 45. bxa6 Bb7 just in time 46. axb7 Nc6 47. b8=B Nd8 and now white could castle if they saw it fit

It

is possible.

Proof

game: 1. Nf3 Nf6 2. Nh4 Nd5 3. Ng6 Nc6 4. Nxh8 Nd4 5. Ng6 Nb3 6. Ne5 Nf4 7. g3 Nxc1 8. Bg2 Ng6 9. Be4 Nh8 10. Rg1 Nb3 11. a3 {let the queen out while preserving as many stalling moves as possible} Nd4 12. Nc3 Ne6 13. Qb1 Ng5 14. Qa2 Ne6 15. Qd5 Rb8 16. Qc6 bxc6 17. Nd5 Ba6 18. Bf3 Bd3 19. Nc4 Bf5 20. Ne5 Bh3 21. Be4 Bf1 {it must be the bishop, a knight would be too slow to reach their final position from here} 22. Rg2 Qc8 23. Nf3 Rb6 24. Rg1 cxd5 25. Rg2 dxe4 26. Rg1 Qa6 27. Rg2 Kd8 28. Rg1 exf3 29. Rg2 Kc8 30. Rg1 Kb7 31. Rg2 Kc6 32. Rg1 Kd5 33. Rg2 Ke4 34. Rg1 Kf5 35. Rg2 Kg4 36. Rg1 Kh3 37. Rg2 Rb3 38. Rg1 Nd8 39. Rg2 fxg2 {from here black needs to finish before white's remaining moves are used up} 40. a4 g1=R 41. a5 Kg2 42. cxb3 Rh1 43. b4 Kg1 44. b5 Bg2 45. bxa6 Bb7 {just in time} 46. axb7 Nc6 47. b8=B Nd8 and now white could castle if they saw it fit

It

is possible.

Proof game:

1. Nf3 Nf6 2. Nh4 Nd5 3. Ng6 Nc6 4. Nxh8 Nd4 5. Ng6 Nb3 6. Ne5 Nf4 7. g3 Nxc1 8. Bg2 Ng6 9. Be4 Nh8 10. Rg1 Nb3 11. a3 let the queen out while preserving as many stalling moves as possible Nd4 12. Nc3 Ne6 13. Qb1 Ng5 14. Qa2 Ne6 15. Qd5 Rb8 16. Qc6 bxc6 17. Nd5 Ba6 18. Bf3 Bd3 19. Nc4 Bf5 20. Ne5 Bh3 21. Be4 Bf1 it must be the bishop, a knight would be too slow to reach their final position from here 22. Rg2 Qc8 23. Nf3 Rb6 24. Rg1 cxd5 25. Rg2 dxe4 26. Rg1 Qa6 27. Rg2 Kd8 28. Rg1 exf3 29. Rg2 Kc8 30. Rg1 Kb7 31. Rg2 Kc6 32. Rg1 Kd5 33. Rg2 Ke4 34. Rg1 Kf5 35. Rg2 Kg4 36. Rg1 Kh3 37. Rg2 Rb3 38. Rg1 Nd8 39. Rg2 fxg2 from here black needs to finish before whites remaining moves are used up 40. a4 g1=R 41. a5 Kg2 42. cxb3 Rh1 43. b4 Kg1 44. b5 Bg2 45. bxa6 Bb7 just in time 46. axb7 Nc6 47. b8=B Nd8 and now white could castle if they saw it fit

It

is possible.

Proof

game: 1. Nf3 Nf6 2. Nh4 Nd5 3. Ng6 Nc6 4. Nxh8 Nd4 5. Ng6 Nb3 6. Ne5 Nf4 7. g3 Nxc1 8. Bg2 Ng6 9. Be4 Nh8 10. Rg1 Nb3 11. a3 let the queen out while preserving as many stalling moves as possible Nd4 12. Nc3 Ne6 13. Qb1 Ng5 14. Qa2 Ne6 15. Qd5 Rb8 16. Qc6 bxc6 17. Nd5 Ba6 18. Bf3 Bd3 19. Nc4 Bf5 20. Ne5 Bh3 21. Be4 Bf1 it must be the bishop, a knight would be too slow to reach their final position from here 22. Rg2 Qc8 23. Nf3 Rb6 24. Rg1 cxd5 25. Rg2 dxe4 26. Rg1 Qa6 27. Rg2 Kd8 28. Rg1 exf3 29. Rg2 Kc8 30. Rg1 Kb7 31. Rg2 Kc6 32. Rg1 Kd5 33. Rg2 Ke4 34. Rg1 Kf5 35. Rg2 Kg4 36. Rg1 Kh3 37. Rg2 Rb3 38. Rg1 Nd8 39. Rg2 fxg2 from here black needs to finish before white's remaining moves are used up 40. a4 g1=R 41. a5 Kg2 42. cxb3 Rh1 43. b4 Kg1 44. b5 Bg2 45. bxa6 Bb7 just in time 46. axb7 Nc6 47. b8=B Nd8 and now white could castle if they saw it fit

It

is possible.

Proof game:

1. Nf3 Nf6 2. Nh4 Nd5 3. Ng6 Nc6 4. Nxh8 Nd4 5. Ng6 Nb3 6. Ne5 Nf4 7. g3 Nxc1 8. Bg2 Ng6 9. Be4 Nh8 10. Rg1 Nb3 11. a3 Nd4 12. Nc3 Ne6 13. Qb1 Ng5 14. Qa2 Ne6 15. Qd5 Rb8 16. Qc6 bxc6 17. Nd5 Ba6 18. Bf3 Bd3 19. Nc4 Bf5 20. Ne5 Bh3 21. Be4 Bf1 22. Rg2 Qc8 23. Nf3 Rb6 24. Rg1 cxd5 25. Rg2 dxe4 26. Rg1 Qa6 27. Rg2 Kd8 28. Rg1 exf3 29. Rg2 Kc8 30. Rg1 Kb7 31. Rg2 Kc6 32. Rg1 Kd5 33. Rg2 Ke4 34. Rg1 Kf5 35. Rg2 Kg4 36. Rg1 Kh3 37. Rg2 Rb3 38. Rg1 Nd8 39. Rg2 fxg2 40. a4 g1=R 41. a5 Kg2 42. cxb3 Rh1 43. b4 Kg1 44. b5 Bg2 45. bxa6 Bb7 46. axb7 Nc6 47. b8=B Nd8

It

is possible.

Proof game:

1. Nf3 Nf6 2. Nh4 Nd5 3. Ng6 Nc6 4. Nxh8 Nd4 5. Ng6 Nb3 6. Ne5 Nf4 7. g3 Nxc1 8. Bg2 Ng6 9. Be4 Nh8 10. Rg1 Nb3 11. a3 let the queen out while preserving as many stalling moves as possible Nd4 12. Nc3 Ne6 13. Qb1 Ng5 14. Qa2 Ne6 15. Qd5 Rb8 16. Qc6 bxc6 17. Nd5 Ba6 18. Bf3 Bd3 19. Nc4 Bf5 20. Ne5 Bh3 21. Be4 Bf1 it must be the bishop, a knight would be too slow to reach their final position from here 22. Rg2 Qc8 23. Nf3 Rb6 24. Rg1 cxd5 25. Rg2 dxe4 26. Rg1 Qa6 27. Rg2 Kd8 28. Rg1 exf3 29. Rg2 Kc8 30. Rg1 Kb7 31. Rg2 Kc6 32. Rg1 Kd5 33. Rg2 Ke4 34. Rg1 Kf5 35. Rg2 Kg4 36. Rg1 Kh3 37. Rg2 Rb3 38. Rg1 Nd8 39. Rg2 fxg2 from here black needs to finish before whites remaining moves are used up 40. a4 g1=R 41. a5 Kg2 42. cxb3 Rh1 43. b4 Kg1 44. b5 Bg2 45. bxa6 Bb7 just in time 46. axb7 Nc6 47. b8=B Nd8 and now white could castle if they saw it fit