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Kantorovich, Steckner and Poghosyan
Diagram 6.04 on page 229, Karsten Muller and Yakov Konoval in their illustrious 2016 chess book, Understanding Rook Endgames

<"This position is very near the borderline between win and draw. Kantorovich had thought in 1989 that it is drawn and this indeed seems to be correct, but for a different reason. Steckner found an improvement in 2003 and until 2011 the position was thought to be won. But then Vardan Poghosyan found a new way to draw. Micawber and Dvoretsky have analyzed it very deeply
(see the 4th edition of Dvoretsky's Endgame Manual).">
Quote by Muller, who has a world-wide reputation as one of the greatest endgame experts.

Also, the above diagram position has been reached many times in top level games:
Tal-Gipslis 1983, Grischuk-Radjabov 2008 and Bacrot-Robson 2011.

From Wikipedia:
2.2 Rook behind enemy passed pawn: usually a draw
2.2.1 New analysis
'https://en.wikipedia.org/wiki/Tarra...

<"Recent theoretical analysis of this position shows that White has a strong maneuver:

1. Advance the pawn to the sixth rank.
2. Move the king towards the queenside.
3. When the black rook takes a kingside pawn, switch the rook to guarding the pawn from the c-file, i.e. Rc7 then advance the pawn to a7.
4. Switch the white rook to the a-file with gain of tempo. Thus Black is forced to sacrifice his rook for the pawn without White having to move his king all the way to a7. These many extra tempos make the difference between winning and drawing or even losing.

Black must play very carefully to draw, rather than the very easy draw that was long thought to be the case.">

(1.Ra8 Kf5 <"Black has an easy draw; however, White has a better plan which wins with precise play:">)
As we shall see later, Black can draw...

E. 1.1
1.Kd4!
<"The f-pawn must be sacrificed because the rook is on a7.">

Black to move. Last: 1.Kd4

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Only one move enables Black to save, which one?

A) 1...Rxf2? 2.Rc7! Ra2 3.a7
(3.Rc6+ <"leads to a draw.">)

3...Kf5 4.Kc4!!
(<"The old analysis was"> 4.Rxf7+, <"leading to a draw.">)

4...Kg4 5.Kb3! (allows for a <bullying check> of Black King) 5...Ra6 6.Rc4+ Kxg3 7.Ra4 Rxa7 8.Rxa7 Kxh4

E. 1.2
White to move. Last: 8...Kxh4
Notice that White has accomplished all 4 steps of above new analysis from Wiki.

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Now 9.Kc2 or Kc3 or Kc4 White wins in 32 moves according to Shredder. The rook will be able to deal with the three pawns but needs the help of the White king first.

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Only one move enables Black to save, which one?

B) 1...Kg2? and now 2.Ra8 transposes to Tal-Gipslis game, here passive play should lose.

<"The point is that the best chance for a successful defense in this sort of position is counterplay on the kingside: creation of a passed pawn or weakening the opponent's position."> Dvoretsky

C) 1...g5!!
<"Poghosyan proves that this draws. This advance could also be played earlier."> Muller

E. 1.3
White to move. Last: 1...g5
See below references 3 & 5 for additional analysis

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2.Kd5!? g4!
<"2. ♔d5 ♙g4!! is very deep, but it does draw."> Muller

3.Kd6 Rd2+ 4.Kc5 Rc2+ 5.Kb5 Rb2+ 6.Kc4 Ra2! 7.Ra8

E. 1.4
Black to move. Last: 7.Ra8

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<"Dvoretsky analyses a loss after 7…Kf5? and Poghosyan gave 7…Ke6? 8.Kb5 Rb2+ 9.Kc6 Rc2+ 10.Kb7 Rb2+ 11.Ka7 Rxf2 Now after 12.Rh8! Rf3 instead of 13.Rxh5? Rxg3= we have 13.Kb6 (or Kb7) Rb3+ 14.Kc6! Ra3 15.Re8+! (15.Kb5? Kd7=) Kf5 16.Kb5 wins If 14…Rc3+ 15.Kb5 Ra3 (15…Rb3+ 16.Ka4 Rb1 17.Rxh5 f5 18.Ka5 wins) 16.Rxh5 Rxg3 17.Rg5 Rb3+ 18.Ka4 Rb8 19.Rxg4 mate in 25

This bullying check of the Black King can be avoided by: 7…Ke7! 8.Kb5 Rb2+ 9.Kc6 Rc2+ 10.Kb7 Rb2+ 11.Ka7 Rxf2 12.Rh8 (12.Rb8 Kd6! draws Poghosyan) Rf3 13.Kb6 Rb3+ 14.Kc5 (or Kc6) Ra3! (14…Rc3+? 15.Kb5 +-) 15.Kb5 Kd7 16.Rxh5 Rxg3 17.Rg5 Kc7 18.Rg8 Rb3+ 19.Ka4 Rb8 20.Rxg4 Rh8 draw">
John Hall

Going back to E. 1.3
White to move. Last: 1...g5

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<"Poghosyan's defense requires the following statements to be true.
a) Black can hold the position after hxg5+
b) Black can hold the positions after he pushes his pawn to g4.">
Micawber

2.hxg5+ Kxg5 3.Kc5

<"These moves including 1...g5! were first seen in Grischuk-Radjabov, 2008. Grischuk continued 3.Kc4= . After 3....Kg4 4.Kb3,Rxf2!= the game ended in a draw as well.">
Micawber

3....Kg4 4.Kb5

E. 1.5
Black to move. Last: 4.Kb5

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<"Black holds the draw by taking on f2 just now.">
Micawber

4...Rxf2 5.Ra8 Rb2+ 6.Kc4 Ra2 7.Rg8+ Kf3 8.Kb5 Rb2+ 9.Kc6 Ra2 10.Kb7 Rb2+ 11.Ka8 f5! 12.Rg5 f4! 13.gxf4 h4! 14.f5 Kf4 15.Rh5 Kg4 16.Rh8 Kxf5
17.Rxh4 Ke6 (Dvoretsky).

References:
Poghosyan, Micawber, and John Hall wrote numerous articles on this theoretical endgame in Chess Publishing like:

1. 'https://www.chesspub.com/cgi-bin/ch...

2. 'https://www.chesspub.com/cgi-bin/ch...

3. 'https://www.chesspub.com/cgi-bin/ch...

4. 'https://www.chesspub.com/cgi-bin/ch...

5. 'https://www.chesspub.com/cgi-bin/ch...

6. 'https://www.chesspub.com/cgi-bin/ch...