Well, this should be semi-simple for a computer, at least. We know with absolute certainty that there are only 20 possible moves for both black and white (pawns and knights) on the first turn. After that, the only possible moves are moving the piece moved again, moving another pawn or knight, or moving a piece freed by the first move.
Knowing this, we can calculate the number of possible board positions after two turns. Luckily this has already been done here:
There are 400 possible chess positions after two ply moves (first ply move for White followed by first ply move for Black).
There are 5,362 possible positions (White’s second ply move) or 8,902 total positions after two ply moves each. There are 71,852 possible positions or 197,742 total positions after four moves.
This is a LOT, but it is possible to at least guess which one was used in the game.
The method I propose is to go though all of the movements of the game after the first four, but starting with the default chessboard as if the game had only just started. If a move occurs which is impossible, this indicates that this move has been made by a piece moved in the first two turns.
Despite this, there could be cases where a pawn moved on white's first turn and a pawn moved on white's third turn can take the same piece. If white takes with the pawn moved on the first turn, there would be no way to spot this, since you would assume the pawn moved on the third turn was used.
Sorry that this is only a half-answer; I will try to find more information to improve this.