My guess is that this problem was composed by Tom Volet. It feels like his style.
White is missing an R and B, and Black is missing a B and 3P. White made two P captures on the Q-side. With only two captures available to Black, it is not possible for any of Black's pawns to be taken there. One of them must have promoted.
The only way to free the NW corner is to pull the c3 pawn back to b2 and let the bK out. This can't be done until Bc1 returns home. The first retractions must obviously be 1.N~ (to release the check) and 1...g1=N. The only way to get Bc1 back on the board is to uncapture on h2, which cannot be done until Pg3 steps back, which can't be done until Bf1 comes home, which can't be done until wK is home, because there is no way for him to get in on the Queenside.
It turns out to be much faster to route the Queen home via g1 than the Q-side. She must get home before both of the Bishops are home and locked in. It is possible to get 26 moves into a retraction that fails only because the Q is still away from home, and Black finally runs out of tempi. Loooong tries like this help make the problem extremely difficult to solve.
1.Na3 g1N 2.Qg1 d4 3.Kf6 d5 4.Kg5 d6 5.Kh4!!
There isn't time to go straight to e3 and uncapture. This gives Black some extra tempi to burn while waiting for the e-pawn to appear.
5...d7 6.Kg5(xPh4) h5 7.Kf4 h6 8.Ke3 h7 9.Kf2(xPe3) e4 10.Ra1 e5 11.Qd1 e6 12.Ke1 e7 13.f2 f3(xRg2) 14.Rg1 f4 15.Bf3 f5 16.Bg2 f6 17.Bf1 f7 18.g2 g3xBh2 19.Nb1 g4 20.Bd6 g5 21.Ba3 g6 22.Bc1 Kb4 23.b2xNc3
The position is unwound, except for wNh8. The only way to get him out is to put a Black Rook on g8, Bishop on f8, and then retract g7-g6. This proves that the first move of Black's King Rook was Rg8.