Well, I'd thought to prove that it was insoluble

- All pawn positions are locked in place. They can neither move nor be taken. - The bottom black bishop is locked into place. He can't move at all without taking a pawn. - the black bishop on a7 can only move back and forth between there and b8. He always consumes an even number of moves to return to origin. - the black bishop on h6 can move to g5, h4, g3, h2, and g1. He always consums an even number of moves to return to origin - The white king is locked into place by the fact that he cannot take a pawn and the fact that he cannot move into check. - The black knight is locked into place. All of his available moves would either involve taking a pawn, or involve putting the white king in check - which would require the white king to take a pawn.  - The only white piece that can move is the rook, and he can only move between d1 and e1. If he attempts to leave the 1 row, he puts his king in check from the black rook, and the black rook cannot escape the pin without first moving to d1, which would force white to make a non-reversible move. he requires an even number of turns to return to origin. the only movable pieces are two of the three black bishops (who can only burn even numbers of turns), the white rook (who can only burn even numbers of turns) and the Black King (who therefore must return to his starting position in an odd number of turns in order to solve this).

...and this is the point where my proof falls apart and I realize that @Braegh has the right of it because

The king can sneak by the white pawns at the top if you move the black bishop all the way down first.

Ah, well.

Well, I'd thought to prove that it was insoluble

Pawns

All pawn positions are locked in place. They can neither move nor be taken.

Black Bishops

The bottom black bishop is locked into place. He can't move at all without taking a pawn. The black bishop on a7 can only move back and forth between there and b8. He always consumes an even number of moves to return to origin. The black bishop on h6 can move to g5, h4, g3, h2, and g1. He always consums an even number of moves to return to origin

White King

The white king is locked into place by the fact that he cannot take a pawn and the fact that he cannot move into check.

Black Knight

The black knight is locked into place. All of his available moves would either involve taking a pawn, or involve putting the white king in check - which would require the white king to take a pawn.

Rooks

- The only white piece that can move is the rook, and he can only move between d1 and e1. If he attempts to leave the 1 row, he puts his king in check from the black rook, and the black rook cannot escape the pin without first moving to d1, which would force white to make a non-reversible move. he requires an even number of turns to return to origin. Thus, the black rook is also locked in place.

Movable pieces

the only movable pieces are two of the three black bishops (who can only burn even numbers of turns), the white rook (who can only burn even numbers of turns) and the Black King (who therefore must return to his starting position in an odd number of turns in order to solve this).

...and this is the point where my proof falls apart and I realize that @Braegh has the right of it because...

The king can sneak by the white pawns at the top if you move the black bishop all the way down first.

Ah, well.