White is missing 4 units (a-pawn, c-pawn, dark-squared bishop, one rook); one of the pawns promoted to a light-squared bishop, since the original was captured at home. Black is missing 3 units (a-pawn, d-pawn, a knight). White made the capture dxe. Also, White's promotion to light-squared bishop must be on c8 or e8 (a8 is blocked by b7, g8 is too far). So either White made two more captures by the c-pawn, or Black's pawn on c6 actually came from d7. Black's pawn on h2 made at least two captures. Current total: Either White made 3 pawn captures and Black 2, or White made 1 and Black 3.
The upper-right cage can be undone in two ways: retracting dxe3, or retracting White's bishop capturing something.
First, look at dxe3. This traps the dark-squared bishop, so Black didn't capture White's dark-squared bishop outside. Since Black also didn't capture the light-squared bishop outside either, this means Black could only have made 2 pawn captures. Thus the latter case is impossible; it's the former case. Black's missing units were all captured by White, and White's missing units were the two original bishops plus the rook and a pawn that got captured.
To recap: White made the capture dxe3, plus two captures used to promote either a-pawn or c-pawn to c8 or e8. Black captured both White's bishops at home, plus hxgxh.
In order for pawns to be meaningfully captured, they must promote. One of White's pawns is the one promoting to a light-squared bishop; the other, and both Black's missing pawns, promoted. In order for Black to promote d-pawn, White must do dxe3, which means the rest of the cage must have been set. But then I can't see how to give White extra moves; it takes at least 3 moves after dxe3 in order for Black to get the d-pawn into position on Rf1 (no other piece is out of the cage).
Now, look at White's bishop capturing something. In this case, it's the former case that's impossible, since White will simply have made too many captures. So White made 1 pawn capture and Black made 3; moreover, White made an additional capture Bxg6.
Black captured all of White's missing units except a pawn, that promoted to a light-squared bishop; this also means both of White's pawns will have to promote either way, since the captures happened on c-, g-, and h-files. This also means Black's a-pawn has to be captured uselessly to free up the way. White captured two of Black's missing units, which can't be the a-pawn, so it's the d-pawn and the knight. But they are on e- and g-files, so Black needs to promote the d-pawn.
Like before, Black needs to promote the d-pawn, so dxe3 must happen first. However, this time it doesn't trap anything, since the cage can be undone as follows: -1... Be4xNg6 -2. Rd1-f1 Bd5-e4 -3. Kf5-g5 Bc4-d5 -4. Ke5-f5 Be6-c4 -5. Nf4-g6 and the whole thing is unlocked.
It's probably me missing something obvious, but it looks like either player can have the move.
EDIT: Oh, of course. It's not d-pawn that promotes; it's Black's c-pawn. Which means Black needs to make an additional capture. This renders the whole second case impossible. But then that means the position is illegal? I'll have to check my work again.
Mate in 1: