Yes, the endgame is winning for White. Take the position after these first moves:
- Qe8+ Kh7 2. Qxe3.
Note here that 2...Rxg2 allows White to win the Rook with the fork: 3.Qe4+
The endgame is now just 2 pawns vs 3 pawns. You assumed that Black can swap off a pair of pawns, but this is easier said than done. If Black pushes their kingside pawns up the board, the king will be exposed and the fortress weakened. White's queen can also help to prevent any unwanted exchanges.
So Black's best strategy to defend is to put the rook on e5, where it prevents White's king from crossing the e-file and reaching the kingside. However, notice how the rook must stay on e5 to accomplish this. If it moves anywhere else on the e-file, it's liable to be forked. Note also that it's not easy for Black to even bring the rook to the e5-square in the first place. But let's assume Black manages.
Therefore, White's winning strategy consists of putting Black in zugzwang. The ideal arrangement of White's pieces:
- King on d6 or d7.
- Queen on f8.
- Pawns on h4 and g4
And then Black's pieces will be as follows:
- King on h7.
- Rook on e5.
- Pawns as they are in the diagram position (moving just hurts Black's defensive chances).
If you set this position up on a board, you'll notice how Black's king can only move to g6. But then White plays Qg8 and Black's king is stuck in a vulnerable position.
Black's only other option is moving the rook somewhere else along the e-file, such as 1...Re1. But then White strikes with 2.g5!, since the Rook is no longer on the 5th rank preventing this push. After 2...fxg5 3.hxg5 hxg5 4. Qf7, Black's rook no longer has a protected square it can sit on along the e-file. This means it won't be able to prevent White's king from crossing the e-file and reaching the kingside much longer.
Once White's king crosses the file, it's easy enough to checkmate Black's king.
EDIT: For Black to move in the starting position:
- On 1...Rxg2 2.Qe8+ Kg5 3.Qxe3+ Kh4 (...Kg6 or ...Kf5 lose the rook to a fork), Black can't really set up a fortress.
- Placing the rook on g5 can eventually lead it to being attacked when White pushes h4 with the help of the queen.
- Meanwhile, putting the rook on e5 leaves the g7-pawn undefended, since the king is stranded on h4. If White takes the g7-pawn, Black's fortress falls apart due to the weak f6-pawn.
- On 1...e2 2.Qg4+ Kh7 3.Qe4+ Kh8 4.g4 Kg8 5.h4, White is going to round up the e2-pawn by bringing the king to b3. If Black's rook moves to d2, White plays Qe3 and Black's rook is unable to stay on the 2nd rank. White's queen then takes the e2-pawn, and see my main answer for how to break the subsequent fortress w/ rook on e5.
- Other first moves for Black don't do anything special.
