I'm new to these and I may be wrong, but I'm gonna say yes it is possible. The points below are a kind of attempt at disproving it that ultimately finds it doesn't seem it can be disproven.:
White's dark-square bishop must be a promoted pawn to be where it is with black's a and c pawns unmoved.
For white's king to be unmoved with white's pawns where they are, black's rook must be a promoted pawn, and specifically originally the b pawn. Furthermore, under the same assumption, it can only have been promoted on g1 or h1.
Black's pawn cannot have been promoted on h1 because it would have captured 6 of white's pieces along its diagonal path, when there are only 5 pieces it can have captured: 2 knights, the queen, the missing rook and the light-square bishop. Additionally, these 5 pieces must have been captured by the black pawn before its promotion.
For black's pawn to have promoted on g1 before moving to h1 without white's king ever moving, there must have been a piece on f1 shielding white's king from the rook until black's king moved to g1. With the aforementioned 5 white pieces having already been captured, this must have been a black piece. Since it can't have checked the king, it must have been a bishop or a knight.
White's dark-square bishop can't have ever moved, as white's b and d pawns are unmoved.
For white's castlability to be intact, the only moves white can have made since the black pawn capturing onto g2 are the advancement of the a and c pawns including at least 1 capture by the c pawn, and the possibility of more captures by these 2 pawns is not to be neglected. For white's queen to make its way into the capture path of the black pawn, one of these white pawns must have already been moved at least once. Therefore maximally 8 moves can have been made by white since the g2 capture.
Since capturing on g2, black will need minimally 4 moves to promote and move the new rook to h1, then move the king to g1.
If the black piece on f1 were a knight, it would need 5 moves to get to its final position, totaling 9 moves for black, so it if this position is reachable where white can still castle, then it must have been a black bishop which was shortly thereafter captured by white's a or c pawn.
Black's light-square bishop needs 2 moves to get b7, which is a good place for it to get captured. This means that 6 moves must have already happened since black's pawn capturing on g2. After white's 6 pawn moves (and the earlier pawn move to free the queen), white's c pawn would already be in c6, blocking the bishop from b7--unless white's c pawn had already captured its way to the a file, that is.
If two of black's pieces, a queen and a rook for example, were lined up for the taking on b5 and a6, then the original c pawn would not be blocking black's bishop and the position is reachable.
Yes it is possible. The points below are a kind of attempt at disproving it that ultimately finds it doesn't seem it can be disproven.
1. White's dark-square bishop must be a promoted pawn to be where it is with black's a and c pawns unmoved.
2. For white's king to be unmoved with white's pawns where they are, black's rook must be a promoted pawn, and specifically originally the b pawn. Furthermore, under the same assumption, it can only have been promoted on g1 or h1.
3. Black's pawn cannot have been promoted on h1 because it would have captured 6 of white's pieces along its diagonal path, when there are only 5 pieces it can have captured: 2 knights, the queen, the missing rook and the light-square bishop. Additionally, these 5 pieces must have been captured by the black pawn before its promotion.
4. For black's pawn to have promoted on g1 before moving to h1 without white's king ever moving, there must have been a piece on f1 shielding white's king from the rook until black's king moved to g1. With the aforementioned 5 white pieces having already been captured, this must have been a black piece. Since it can't have checked the king, it must have been a bishop or a knight.
5. White's dark-square bishop can't have ever moved, as white's b and d pawns are unmoved.
6. For white's castlability to be intact, the only moves white can have made since the black pawn capturing onto g2 are the advancement of the a and c pawns including at least 1 capture by the c pawn, and the possibility of more captures by these 2 pawns is not to be neglected. For white's queen to make its way into the capture path of the black pawn, one of these white pawns must have already been moved at least once. Therefore maximally 8 moves can have been made by white since the g2 capture.
7. Since capturing on g2, black will need minimally 4 moves to promote and move the new rook to h1, then move the king to g1.
8. If the black piece on f1 were a knight, it would need 5 moves to get to its final position, totaling 9 moves for black, so it if this position is reachable where white can still castle, then it must have been a black bishop which was shortly thereafter captured by white's a or c pawn.
9. Black's light-square bishop needs 2 moves to get b7, which is a good place for it to get captured. This means that 6 moves must have already happened since black's pawn capturing on g2. After white's 6 pawn moves (and the earlier pawn move to free the queen), white's c pawn would already be in c6, blocking the bishop from b7--unless white's c pawn had already captured its way to the a file, that is.
10. If two of black's pieces, a queen and a rook for example, were lined up for the taking on b5 and a6, then the original c pawn would not be blocking black's bishop and the position is reachable.
Thinking about further, while 1-8 are pretty invariable as far as I can tell, there seem to be a lot of ways it could shake out with the pawns. For instance, the a pawn might actually be the one that captures the bishop on b7, while the c pawn captures to the a file, or maybe the bishop doesn't even get captured on b7.
While 1-8 are pretty invariable as far as I can tell, there seem to be a lot of ways it could shake out with the pawns. For instance, the a pawn might actually be the one that captures the bishop on b7, while the c pawn captures to the a file, or maybe the bishop doesn't even get captured on b7.