EDITED after ~15 hours.
As Amaud Mortier points out in their answer, without anything "weird" going on, White is losing. The only remaining hope for White to draw is to claim a draw by threefold repetition or the fifty-move rule. But it's impossible to prove the former is possible, and the latter seems to be outright impossible to prove too, since Amaud has provided a game that goes nowhere near the 50-move limit that ends in this position...
However, since this is a retrograde-analysis puzzle:
Castling is always assumed to be possible unless it's provably impossible. With that in mind, what happens if White plays O-O-O? Perhaps, by forcing castling to have been possible, we can force the game to have taken 49 non-pawn moves before this position, and immediately claim a draw before Qa1#?
Let's examine how the position in the diagram above arose (incomplete):
We first note that White is missing a knight, a queen, and a pawn. We also note that Black is missing nothing. Black has also made at least three pawn captures (by the pawns on b4, b6, f5, and g6), so Black has made exactly three pawn captures, and White has made no pawn captures since Black is missing nothing. This accounts for all the pieces on the board.
The next question is, how did Black's bishop get to g8? Since Black has not promoted any pieces, as all eight of their pawns are on the board, the bishop must have come in via Bh7-Bg8, and then a pawn must have gotten to g6. The only place for such a pawn to have come from without blocking the bishop first is g7, so the pawn on f5 must have come from h7. So the move sequence must have been, in some order, h7xg6, g6xf5, Bh5, Bg6, Bh7, Bg8, g6.
How did White's bishop get to b8? Since White has not captured, any promotion of the missing a-pawn must have been on a8, so the bishop on b8 cannot be a promoted bishop; it must be the original bishop from c1. By a similar logic to the above, the moves in this area must have been Ba7, Bb8, b6. So the pawn at b4 has made Black's third capture. Further, this move sequence must have occurred before Black moved their light-square bishop.
Note also that in both corners on the 8th rank, there are hemmed-in knights. These must have gotten there before b6 and g6 were pushed. So we must have had, in roughly this order: Ba7, Bb8, Na1, b6, h7xg6, g6xf5, Bh5, Bg6, Bh7, Bg8, Nh8, g6.
For the white pawn at b5 to get where it is, it must have moved to b5 first, then Black must have captured at b4. Now, this means that although White's a-pawn was missing, it cannot have been captured on the a-file. Therefore, it must have promoted.. For it to have promoted, the a-file had to have been clear, so it had to have promoted after the capture at b4 but before the knight moved to a8 (and thus before the move b6); it can only therefore have been captured on f5 or g6 by the h-pawn, if at all.
White's rook, in order to have escaped to be captured or to end up on f8, must have passed through f1 and f3 (since we are assuming that White can castle), so f4 must have been played before this; similarly, White's bishop must have passed through e2, so e3 must have been played before this.
Which pawn moved last, and was it before or after the last capture? The last capture was made by either the b4 or f5 pawn, after which g6 was played, the last pawn move must have occurred after the last capture. It cannot have been to a8, b4, b5, b6, or f5 from the reasons above, as these moves all occurred before g6. Other than the stationary pawns that have never moved, the only remaining candidates are the moves g6, e3, f4, and h3.
Let's turn to the black bishop at g3. It can only have moved where it is after Black played g6. However, the whole configuration on the right-hand-side of the board seems pretty locked-up; Black's knight has nowhere to go, Black's rooks can't move without retro-checking White, and even if they weren't there, Black couldn't move the knight without retro-checking White either. We can only conclude that Black's knight has to have gotten to its current position via either f5 or g6.
What remains to be proven:
That if White can castle, there must have been 49 moves prior to the current move.
That there is a game that ends in this position where White can indeed castle (i.e. that one cannot disprove that White can castle).