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Final position:

Lower bound for shortest proof game :

Note
1. that each black N must have entered its respective corner before the nearby P can move.
2. that the Ks cannot enter their final destinations when the Bs are already there. But something must shield the Ks from the Rs. Therefore the Rs must be one square further away and the white Ns be in between Ks and Rs. Once the Ks are in the Bs can follow and then the white Ns can move on and the Rs take their final positions.

with that in mind we find that
overall white moves are limiting
on top of that there is a "tempo" constraint in the beginning becaus black has to make 9 N plies before the two Ps can move
we'll need at the very least the following white plies:

11 plies to move the K from e1 via b2 and h6 to g8
1 ply to move the P
4 plies for Rs: Rh1-g1 and Ra1-b1-e1-d1
17 plies for Ns: We need to move a N to d1 but Nb1-c3-d1 does not work because the of the following necessary sequence of evens: N has to leave b1 to allow Ra1-b1 which is required to allow a black N to a1 which must happen before Pb2-b3 which is required to allow the white K out which must move through c3 or waste two plies. But if the N moves from c3 to d1 to make room for the K it will prevent the black K to enter via b2. Therefore we know that getting a white N to d1,e8,h1 and a8 will take at least four plies each, plus one because of parity.
5 plies to get the Q from d1 to h6 (Qd1-c1-b2-g7-h6 is not possible because the Q has to leave b2 before the white K can move out but may not enter g7 before the black K has moved out)
4 plies to get the B out and back
Total: 42 white plies or 41 1/2 moves

Proof game (still one move too long, B makes one extra move):

1.Na3 Nc6 2.Nb5 Na5 3.Rb1 Nb3 4.Nh3 Na1 5.b3 Nf6 6.Ba3 Nh5 7.Bd6 Nf4 8.Qc1 Ng6 9.Qa3 Rg8 10.Kd1 Nh8 11.Kc1 g6 12.Kb2 Rb8 13.Kc3 Bg7+ 14.Kd3 Bc3 15.Ke3 Kf8 16.Kf4 Kg7 17.Rg1 Kf6 18.Be5+ Ke6 19.Kg5 Qf8 20.Nd6 Qg7 21.Qb4 Qf6+ 22.Kh6 Rd8 23.Ne8 Kd5 24.Qh4 Kc5 25.Nf4 Kb4 26.Re1 Qa6 27.Kg7 Ka3 28.Nd5 Kb2 29.Ne3 Qa3 30.Nd1+ Kb1 31.Kg8 Bb4 32.Bb2 Bc3 33.Bc1 Bg7 34.Qh6 Bf8 35.Nf6 Re8 36.Nh5 Qa4 37.Ng3 Qa5 38.Nh1 Qa6 39.Ne3 Qb6 40.Nd5 Qc6 41.Nb6 Qd6 42.Na8 Qa3 43.Rd1 *

Final position:

Lower bound for shortest proof game :

Note
1. that each black N must have entered its respective corner before the nearby P can move.
2. that the Ks cannot enter their final destinations when the Bs are already there. But something must shield the Ks from the Rs. Therefore the Rs must be one square further away and the white Ns be in between Ks and Rs. Once the Ks are in the Bs can follow and then the white Ns can move on and the Rs take their final positions.

with that in mind we find that
overall white moves are limiting
on top of that there is a "tempo" constraint in the beginning becaus black has to make 9 N plies before the two Ps can move
we'll need at the very least the following white plies:

11 plies to move the K from e1 via b2 and h6 to g8
1 ply to move the P
4 plies for Rs: Rh1-g1 and Ra1-b1-e1-d1
17 plies for Ns: We need to move a N to d1 but Nb1-c3-d1 does not work because the of the following necessary sequence of evens: N has to leave b1 to allow Ra1-b1 which is required to allow a black N to a1 which must happen before Pb2-b3 which is required to allow the white K out which must move through c3 or waste two plies. But if the N moves from c3 to d1 to make room for the K it will prevent the black K to enter via b2. Therefore we know that getting a white N to d1,e8,h1 and a8 will take at least four plies each, plus one because of parity.
5 plies to get the Q from d1 to h6 (Qd1-c1-b2-g7-h6 is not possible because the Q has to leave b2 before the white K can move out but may not enter g7 before the black K has moved out)
4 plies to get the B out (to let in the black K) and back
Total: 42 white plies or 41 1/2 moves

Proof game with minimal number of moves:

1.Nc3 Na6 2.Nf3 Nc5 3.Rb1 Nb3 4.Na4 Na1 5.b3 Nh6 6.Ba3 Ng4 7.Bd6 Ne5 8.Qc1 Rg8 9.Qa3 Ng6 10.Kd1 Nh8 11.Kc1 g6 12.Kb2 Bh6 13.Kc3 Kf8 14.Kd4 Be3+ 15.Ke4 Kg7 16.Qc5 Kf6 17.Qh5 Bc5 18.Kf4 Ke6 19.Kg5 Qf8 20.Rg1 Qg7 21.Re1 Qd4 22.Kh6 Rd8 23.Ng5+ Kd5 24.Ne4+ Kc6 25.Nf6 Kb5 26.Kg7 Kb4 27.Qh6 Qc4 28.Ne8 Ka3 29.Nc3 Kb2 30.Nd1+ Kb1 31.Kg8 Bd4 32.Ba3 Bg7 33.Bc1 Bf8 34.Nf6 Qa4 35.Ne4 Qa3 36.Ng3 Re8 37.Nh1 Rb8 38.Ne3 Qa6 39.Nd5 Qd6 40.Nb6 Qb4 41.Rd1 Qa3 42.Na8 *

Final position:

Lower bound for shortest proof game :

Note 1. that each black N must have entered its respective corner before the nearby P can move. 2. that the Ks cannot enter their final destinations when the Bs are already there. But something must shield the Ks from the Rs. Therefore the Rs must be one square further away and the white Ns be in between Ks and Rs. Once the Ks are in the Bs can follow and then the white Ns can move on and the Rs take their final positions.

with that in mind we find that we'll need at the very least:

23 plies for Ks 8 plies for Rs 2 plies for Ps 9 plies for black Ns 15 plies for white Ns 10 plies for Qs 10 plies for Bs
but I don't think this is achievable

Proof game (not sure it's optimal, though) UPDATE: shorter version

1.Na3 Na6 2.Nb5 Nc5 3.Rb1 Nb3 4.Nh3 Na1 5.b3 Nh6 6.Ba3 Nf5 7.Bd6 Nh4 8.Qc1 Ng6 9.Qa3 Rg8 10.Kd1 Nh8 11.Kc1 g6 12.Kb2 Rb8 13.Kc3 Bg7+ 14.Kd3 Bc3 15.Qa6 Kf8 16.Ke4 Kg7 17.Bg3 Ba5 18.Nd6 Kf6 19.Kf4 Qf8 20.Ng5 Qh6 21.Re1 Qh3 22.Nge4+ Ke6 23.Kg5 Qg4+ 24.Kh6 Rd8 25.Kg7 Kd5 26.Nc3+ Kc5 27.Ne8 Kb4 28.Kg8 Ka3 29.Bd6+ Kb2 30.Nd1+ Kb1 31.Ba3 Bc3 32.Bc1 Bg7 33.Qa3 Qh5 34.Ne3 Bf8 35.Nf6 Qh6 36.Nfd5 Re8 37.Rd1 Qh4 38.Nb6 Qh5 39.Nf5 Qh4 40.Ng3 Qh5 41.Rg1 Qh4 42.Nh1 Qh6 43.Na8 *

Final position:

Lower bound for shortest proof game :

Note
1. that each black N must have entered its respective corner before the nearby P can move.
2. that the Ks cannot enter their final destinations when the Bs are already there. But something must shield the Ks from the Rs. Therefore the Rs must be one square further away and the white Ns be in between Ks and Rs. Once the Ks are in the Bs can follow and then the white Ns can move on and the Rs take their final positions.

with that in mind we find that
overall white moves are limiting
on top of that there is a "tempo" constraint in the beginning becaus black has to make 9 N plies before the two Ps can move
we'll need at the very least the following white plies:

11 plies to move the K from e1 via b2 and h6 to g8
1 ply to move the P
4 plies for Rs: Rh1-g1 and Ra1-b1-e1-d1
17 plies for Ns: We need to move a N to d1 but Nb1-c3-d1 does not work because the of the following necessary sequence of evens: N has to leave b1 to allow Ra1-b1 which is required to allow a black N to a1 which must happen before Pb2-b3 which is required to allow the white K out which must move through c3 or waste two plies. But if the N moves from c3 to d1 to make room for the K it will prevent the black K to enter via b2. Therefore we know that getting a white N to d1,e8,h1 and a8 will take at least four plies each, plus one because of parity.
5 plies to get the Q from d1 to h6 (Qd1-c1-b2-g7-h6 is not possible because the Q has to leave b2 before the white K can move out but may not enter g7 before the black K has moved out)
4 plies to get the B out and back
Total: 42 white plies or 41 1/2 moves

Proof game (still one move too long, B makes one extra move):

1.Na3 Nc6 2.Nb5 Na5 3.Rb1 Nb3 4.Nh3 Na1 5.b3 Nf6 6.Ba3 Nh5 7.Bd6 Nf4 8.Qc1 Ng6 9.Qa3 Rg8 10.Kd1 Nh8 11.Kc1 g6 12.Kb2 Rb8 13.Kc3 Bg7+ 14.Kd3 Bc3 15.Ke3 Kf8 16.Kf4 Kg7 17.Rg1 Kf6 18.Be5+ Ke6 19.Kg5 Qf8 20.Nd6 Qg7 21.Qb4 Qf6+ 22.Kh6 Rd8 23.Ne8 Kd5 24.Qh4 Kc5 25.Nf4 Kb4 26.Re1 Qa6 27.Kg7 Ka3 28.Nd5 Kb2 29.Ne3 Qa3 30.Nd1+ Kb1 31.Kg8 Bb4 32.Bb2 Bc3 33.Bc1 Bg7 34.Qh6 Bf8 35.Nf6 Re8 36.Nh5 Qa4 37.Ng3 Qa5 38.Nh1 Qa6 39.Ne3 Qb6 40.Nd5 Qc6 41.Nb6 Qd6 42.Na8 Qa3 43.Rd1 *

Final position:

Lower bound for shortest proof game :

Note 1. that each black N must have entered its respective corner before the nearby P can move. 2. that the Ks cannot enter their final destinations when the Bs are already there. But something must shield the Ks from the Rs. Therefore the Rs must be one square further away and the white Ns be in between Ks and Rs. Once the Ks are in the Bs can follow and then the white Ns can move on and the Rs take their final positions.

with that in mind we find that we'll need at the very least:

23 plies for Ks 8 plies for Rs 2 plies for Ps 9 plies for black Ns 15 plies for white Ns 10 plies for Qs 10 plies for Bs
but I don't think this is achievable

Proof game (not sure it's optimal, though).

1. Nf3 Nc6 2. Ne5 Nd4 3. Nc3 Nb3 4. Rb1 Na1 5. b3 Nf6 6. Ba3 Nd5 7. Bd6 Nf4 8. Qc1 Ng6 9. Qa3 Rg8 10. Kd1 Nh8 11. Kc1 g6 12. Kb2 Bh6 13. Re1 Kf8 14. Nd1 Kg7 15. Kc3 Qf8 16. Nc4 Kf6 17. Bg3 Be3 18. Qa6+ Bb6 19. Kd3 Qh6 20. Ke4 Ke6 21. Nd6 Rd8 22. Ne8 Qh3 23. Kf4 Kd5 24. Kg5 Kc5 25. Ne3 Qe6 26. Kh6 Kb4 27. Qd3 Ka3 28. Kg7 Kb2 29. Kg8 Bd4 30. Nd1+ Kb1 31. Bd6 Bg7 32. Ba3 Bf8 33. Bc1 Rb8 34. Rg1 Qh3 35. Qa6 Qh6 36. Qa3 Qh5 37. Nd6 Qh6 38. Nc4 Qh5 39. Nb6 Qh6 40. Na8 Re8 41. Ne3 Qh5 42. Nf5 Qh6 43. Ng3 Qh5 44. Nh1 Qh6 45. Rd1

Final position:

Lower bound for shortest proof game :

Note 1. that each black N must have entered its respective corner before the nearby P can move. 2. that the Ks cannot enter their final destinations when the Bs are already there. But something must shield the Ks from the Rs. Therefore the Rs must be one square further away and the white Ns be in between Ks and Rs. Once the Ks are in the Bs can follow and then the white Ns can move on and the Rs take their final positions.

with that in mind we find that we'll need at the very least:

23 plies for Ks 8 plies for Rs 2 plies for Ps 9 plies for black Ns 15 plies for white Ns 10 plies for Qs 10 plies for Bs
but I don't think this is achievable

Proof game (not sure it's optimal, though) UPDATE: shorter version

1.Na3 Na6 2.Nb5 Nc5 3.Rb1 Nb3 4.Nh3 Na1 5.b3 Nh6 6.Ba3 Nf5 7.Bd6 Nh4 8.Qc1 Ng6 9.Qa3 Rg8 10.Kd1 Nh8 11.Kc1 g6 12.Kb2 Rb8 13.Kc3 Bg7+ 14.Kd3 Bc3 15.Qa6 Kf8 16.Ke4 Kg7 17.Bg3 Ba5 18.Nd6 Kf6 19.Kf4 Qf8 20.Ng5 Qh6 21.Re1 Qh3 22.Nge4+ Ke6 23.Kg5 Qg4+ 24.Kh6 Rd8 25.Kg7 Kd5 26.Nc3+ Kc5 27.Ne8 Kb4 28.Kg8 Ka3 29.Bd6+ Kb2 30.Nd1+ Kb1 31.Ba3 Bc3 32.Bc1 Bg7 33.Qa3 Qh5 34.Ne3 Bf8 35.Nf6 Qh6 36.Nfd5 Re8 37.Rd1 Qh4 38.Nb6 Qh5 39.Nf5 Qh4 40.Ng3 Qh5 41.Rg1 Qh4 42.Nh1 Qh6 43.Na8 *