A difficult rook ending I need help. In the below endgame, Stockfish and Fat Fritz are optimistic on Whit's winning chances , but Mueller disagrees. What happens with perfect play?
[FEN &quot;R7/6p1/P3p1k1/7p/8/5PPP/r7/4K3 b - - 0 1&quot;]
This answer is intended to expand upon my comments, and give some concrete lines. Let's discuss the position as I (mistakenly) first thought it was presented:
[fen &quot;R7/6p1/P3p1k1/7p/8/6PP/r7/4K3 w - - 0 1&quot;]

This position is a draw, no matter who is to move. If White doesn't play a7, then Black's rook will have the option to pick off one of White's pawns on the kingside. So we may as well assume that White is to move, and plays 1.a7. After this, Black will be able to play 1...Kf7, to guard the pawn on e6. Let's examine the resulting position:
[fen &quot;R7/P4kp1/4p3/7p/8/6PP/r7/4K3 w - - 0 1&quot;]

Note that this position would have been losing for Black if it wasn't for the pawn on g7; if there were no pawn on g7, then White would win by replying with 2.Rh8 followed by 2...Rxa7 and 3.Rh7+, picking up Black's rook. But this plan is not possible at the current moment. So White has two different plans to try out: 1) march the king to b5 to free the rook on a8, or 2) create a passed pawn on the h-file, and try to remove the pawn on g7 by means of playing h6 at some point.
The first plan always fails, since Black will keep their rook on the a-file, and when White's king reaches the 4th rank, Black will place their rook on either a1 or a2. Then, if White plays Kb5, Black will give check with ...Rb1+, chasing the white king around until it decides to come back to the fourth rank, after which Black will once again play ...Ra1/Ra2.
The second plan also fails, since if White manages to create the passer on the h-file, and manages to play h6, then Black doesn't have to respond with ...gxh6, but can instead play ...Kg6 to contain the h-pawn. So the conclusion is that this position is indeed a draw.
Let's move on to the actual position you were asking about.
[fen &quot;R7/6p1/P3p1k1/7p/8/5PPP/r7/4K3 w - - 0 1&quot;]

This is also a draw, although not as trivial (I originally assumed that White was winning here, but I missed a simple idea that maintains the balance). Here White has an extra f-pawn, but this doesn't really make a big difference in the long run. Black will use the exact same strategy as before, putting the king on f7 and shuffling the rook along the a-file, but here Black has to be a bit more careful with the winning plan that White tries. As before, the plan of bringing the king to b5 for White is not going to work. Neither is an attack on the pawn on e6 very likely to succeed as long as Black's king is left on f7. So White only has two feasible winning plans:

create a passed pawn on the f-file, and run it to f6 to remove the pawn on g7, or
create a passed pawn on the h-file instead, with the same intention.

The first plan is not going to work well, since White will have to at the very least trade the g-pawn for Black's h-pawn, and then trade another pawn for Black's e6 pawn. At the end of the line, we will end up in the following scenario if White tries winning plan 1:
[fen &quot;R7/P4kp1/5P2/8/8/8/r7/4K3 w - - 0 1&quot;]

Note that if Black just sits on the position and refuses to take on f6, then White can't really do anything to make any sort of progress (this is what I missed initially); taking on g7 just reduces to a trivial draw, and marching the king to the queenside is a lost cause, as always.
Plan 2 is actually the more dangerous one, and here Black needs a bit of precision to stay out of danger:
[fen &quot;R7/6p1/P3p1k1/7p/8/5PPP/r7/4K3 w - - 0 1&quot;]

1.a7 Kf7 2.g4 hxg4 3.fxg4 Ra3 4.h4 Kg6! {This move is absolutely necessary if Black wants to draw. The point is that White was threatening the simple idea of h5-g5-g6+, after which Black is completely lost. Black needs their king on h7 to face this idea.} 5.h5+ Kh7! {Black can also place the king on g5, but then White still has some tricks to look out for. This drawing idea is by far the simplest one.} 6.g5 {with the intention of playing g6+} 6...g6! {This is the point of transferring Black's king to h7; if it were on f7 instead, then White could simply play h6-h7-h8Q, with a completely winning position.} 7.hxg6+ (7.h6= {and Black draws by shuffling the rook along the a-file, chasing White's king away whenever it tries to support the pawn on a7.}) Kg7!= {Black draws by moving the rook along the a-file, and chasing White's king away whenever it tries to support the pawn on a7.}