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First note that

no matter which pawn we move, at least one bishop will be trapped on either side, so we can move at most six non-pawn pieces per side to the other side – the trapped bishop also traps the corresponding rook, albeit in a two-cell jail.

With that in mind, here is a position that can be proved to take

30 moves (60 plies): QNB1Rbrn/ppp1pppp/8/K2p4/4P2k/8/PPPP1PPP/NRBr1bnq

Given the pawn structure, the minimum numbers of plies needed to accomplish the necessary piece moves are:
Free rooks: 7 (white) + 7 (black) Knights and trapped rooks: 10 + 10
(Free) bishops: 3 + 3
Kings: 5 + 5
Pawns: 1 + 1
Queens: 3 + 4
Total: 59

However, if the proof game had 59 plies, black would have one more ply than white, which is impossible. Hence the proof game has 60 plies, and this is achievable:
1. e4 d5 2. Nc3 Nf6 3. Rb1 Rg8 4. Na4 Nh5 5. Nc5 Nf4 6. Nb3 Ng6 7. Na1 Nh8 8. Ke2 Kd7 9. Kd3 Ke6 10. Kc3 Kf6 11. Kb4 Kg5 12. Ka5 Kh4 13. Ne2 Nd7 14. Nd4 Ne5 15. Qe2 Qd7 16. Qb5 Qg4 17. Qe8 Qd1 18. Bb5 Bg4 19. Re1 Rd8 20. Re3 Rd6 21. Rf3 Rc6 22. Rf6 Rc3 23. Rd6 Re3 24. Qa8 Qh1 25. Rd8 Re1 26. Re8 Rd1 27. Nc6 Nf3 28. Nb8 Ng1 29. Bd7 Be2 30. Bc8 Bf1

First note that

no matter which pawn we move, at least one bishop will be trapped on either side, so we can move at most six non-pawn pieces per side to the other side – the trapped bishop also traps the corresponding rook, albeit in a two-cell jail.

With that in mind, here is a position that can be proved to take

30 moves (60 plies): QNB1Rbrn/ppp1pppp/8/K2p4/4P2k/8/PPPP1PPP/NRBr1bnq

Given the pawn structure, the minimum numbers of plies needed to accomplish the necessary piece moves are:
Free rooks: 7 (white) + 7 (black) 
Knights and trapped rooks: 10 + 10
(Free) bishops: 3 + 3
Kings: 5 + 5
Pawns: 1 + 1
Queens: 3 + 4
Total: 59

However, if the proof game had 59 plies, black would have one more ply than white, which is impossible. Hence the proof game has 60 plies, and this is achievable:
1. e4 d5 2. Nc3 Nf6 3. Rb1 Rg8 4. Na4 Nh5 5. Nc5 Nf4 6. Nb3 Ng6 7. Na1 Nh8 8. Ke2 Kd7 9. Kd3 Ke6 10. Kc3 Kf6 11. Kb4 Kg5 12. Ka5 Kh4 13. Ne2 Nd7 14. Nd4 Ne5 15. Qe2 Qd7 16. Qb5 Qg4 17. Qe8 Qd1 18. Bb5 Bg4 19. Re1 Rd8 20. Re3 Rd6 21. Rf3 Rc6 22. Rf6 Rc3 23. Rd6 Re3 24. Qa8 Qh1 25. Rd8 Re1 26. Re8 Rd1 27. Nc6 Nf3 28. Nb8 Ng1 29. Bd7 Be2 30. Bc8 Bf1

First note that

no matter which pawn we move, at least one bishop will be trapped on either side, so we can move at most five non-pawn pieces per side to the other side – the trapped bishop also traps the corresponding rook, albeit in a two-cell jail.

With that in mind, here is a position that can be proved to take

30 moves (60 plies): QNB1Rbrn/ppp1pppp/8/K2p4/4P2k/8/PPPP1PPP/NRBr1bnq

Given the pawn structure, the minimum numbers of plies needed to accomplish the necessary piece moves are:
Free rooks: 7 (white) + 7 (black) Knights and trapped rooks: 10 + 10
(Free) bishops: 3 + 3
Kings: 5 + 5
Pawns: 1 + 1
Queens: 3 + 4
Total: 59

However, if the proof game had 59 plies, black would have one more ply than white, which is impossible. Hence the proof game has 60 plies, and this is achievable:
1. e4 d5 2. Nc3 Nf6 3. Rb1 Rg8 4. Na4 Nh5 5. Nc5 Nf4 6. Nb3 Ng6 7. Na1 Nh8 8. Ke2 Kd7 9. Kd3 Ke6 10. Kc3 Kf6 11. Kb4 Kg5 12. Ka5 Kh4 13. Ne2 Nd7 14. Nd4 Ne5 15. Qe2 Qd7 16. Qb5 Qg4 17. Qe8 Qd1 18. Bb5 Bg4 19. Re1 Rd8 20. Re3 Rd6 21. Rf3 Rc6 22. Rf6 Rc3 23. Rd6 Re3 24. Qa8 Qh1 25. Rd8 Re1 26. Re8 Rd1 27. Nc6 Nf3 28. Nb8 Ng1 29. Bd7 Be2 30. Bc8 Bf1

First note that

no matter which pawn we move, at least one bishop will be trapped on either side, so we can move at most six non-pawn pieces per side to the other side – the trapped bishop also traps the corresponding rook, albeit in a two-cell jail.

With that in mind, here is a position that can be proved to take

30 moves (60 plies): QNB1Rbrn/ppp1pppp/8/K2p4/4P2k/8/PPPP1PPP/NRBr1bnq

Given the pawn structure, the minimum numbers of plies needed to accomplish the necessary piece moves are:
Free rooks: 7 (white) + 7 (black) Knights and trapped rooks: 10 + 10
(Free) bishops: 3 + 3
Kings: 5 + 5
Pawns: 1 + 1
Queens: 3 + 4
Total: 59

However, if the proof game had 59 plies, black would have one more ply than white, which is impossible. Hence the proof game has 60 plies, and this is achievable:
1. e4 d5 2. Nc3 Nf6 3. Rb1 Rg8 4. Na4 Nh5 5. Nc5 Nf4 6. Nb3 Ng6 7. Na1 Nh8 8. Ke2 Kd7 9. Kd3 Ke6 10. Kc3 Kf6 11. Kb4 Kg5 12. Ka5 Kh4 13. Ne2 Nd7 14. Nd4 Ne5 15. Qe2 Qd7 16. Qb5 Qg4 17. Qe8 Qd1 18. Bb5 Bg4 19. Re1 Rd8 20. Re3 Rd6 21. Rf3 Rc6 22. Rf6 Rc3 23. Rd6 Re3 24. Qa8 Qh1 25. Rd8 Re1 26. Re8 Rd1 27. Nc6 Nf3 28. Nb8 Ng1 29. Bd7 Be2 30. Bc8 Bf1

First note that

no matter which pawn we move, at least one bishop will be trapped on either side, so we can move at most six pieces per side to the other side – the trapped bishop also traps the corresponding rook, albeit in a two-cell jail. Furthermore, it is not possible for a king to get to the opposite rank without putting themselves in check from pawns, so the position given in OP's "hint" is illegal (unreachable).

With that in mind, here is a position that can be proved to take

30 moves (60 plies): QNB1Rbrn/ppp1pppp/8/K2p4/4P2k/8/PPPP1PPP/NRBr1bnq

Given the pawn structure, the minimum numbers of plies needed to accomplish the necessary piece moves are:
Free rooks: 7 (white) + 7 (black) Knights and trapped rooks: 10 + 10
(Free) bishops: 3 + 3
Kings: 5 + 5
Pawns: 1 + 1
Queens: 3 + 4
Total: 59

However, if the proof game had 59 plies, black would have one more ply than white, which is impossible. Hence the proof game has 60 plies, and this is achievable:
1. e4 d5 2. Nc3 Nf6 3. Rb1 Rg8 4. Na4 Nh5 5. Nc5 Nf4 6. Nb3 Ng6 7. Na1 Nh8 8. Ke2 Kd7 9. Kd3 Ke6 10. Kc3 Kf6 11. Kb4 Kg5 12. Ka5 Kh4 13. Ne2 Nd7 14. Nd4 Ne5 15. Qe2 Qd7 16. Qb5 Qg4 17. Qe8 Qd1 18. Bb5 Bg4 19. Re1 Rd8 20. Re3 Rd6 21. Rf3 Rc6 22. Rf6 Rc3 23. Rd6 Re3 24. Qa8 Qh1 25. Rd8 Re1 26. Re8 Rd1 27. Nc6 Nf3 28. Nb8 Ng1 29. Bd7 Be2 30. Bc8 Bf1

First note that

no matter which pawn we move, at least one bishop will be trapped on either side, so we can move at most five non-pawn pieces per side to the other side – the trapped bishop also traps the corresponding rook, albeit in a two-cell jail.

With that in mind, here is a position that can be proved to take

30 moves (60 plies): QNB1Rbrn/ppp1pppp/8/K2p4/4P2k/8/PPPP1PPP/NRBr1bnq

Given the pawn structure, the minimum numbers of plies needed to accomplish the necessary piece moves are:
Free rooks: 7 (white) + 7 (black) Knights and trapped rooks: 10 + 10
(Free) bishops: 3 + 3
Kings: 5 + 5
Pawns: 1 + 1
Queens: 3 + 4
Total: 59

However, if the proof game had 59 plies, black would have one more ply than white, which is impossible. Hence the proof game has 60 plies, and this is achievable:
1. e4 d5 2. Nc3 Nf6 3. Rb1 Rg8 4. Na4 Nh5 5. Nc5 Nf4 6. Nb3 Ng6 7. Na1 Nh8 8. Ke2 Kd7 9. Kd3 Ke6 10. Kc3 Kf6 11. Kb4 Kg5 12. Ka5 Kh4 13. Ne2 Nd7 14. Nd4 Ne5 15. Qe2 Qd7 16. Qb5 Qg4 17. Qe8 Qd1 18. Bb5 Bg4 19. Re1 Rd8 20. Re3 Rd6 21. Rf3 Rc6 22. Rf6 Rc3 23. Rd6 Re3 24. Qa8 Qh1 25. Rd8 Re1 26. Re8 Rd1 27. Nc6 Nf3 28. Nb8 Ng1 29. Bd7 Be2 30. Bc8 Bf1