Tablebase 1 - Levenfish & Smyslov ½ (Again)

In Levenfish's Rook Endings, I discussed the book Rook Endings, co-authored by Levenfish & Smyslov, and in Tablebase 1 - Levenfish & Smyslov ½, I gave an atypical error in analysis from the book. No one could reasonably expect that to be the only error and I recently spotted another.

The position in the diagram is from a section on Rook plus a-/h-Pawns vs. Rook. Two Rook Pawns are the sort of position that good players steer for when they realize they are losing. The authors first analyzed the diagrammed position with the g-Pawn shifted to the h-file, showing that it was drawn. Then they analyzed the diagrammed position, showing that it was won for White. Although their conclusion was correct, the analysis was not.

The authors started by giving a variation that White should avoid: 1...Rc4 2.a5 Rc5 3.a6 Rc6 4.g5 Kh7 5.Kg4 Rd6 6.Kh5. Now Black plays the surprising 6...Rh6+! 'with a draw or stalemate'. The draw occurs on 7.Kg4, when 7...Rd6 repeats the position after 5...Rd6, while 7.gxh6 is stalemate. So where is the error?

Levenfish & Smyslov no.130a
Black to Move

In fact, after 5.Kg4 Rd6, the move 6.Kh5 is not the best, precisely because it allows the (pseudo) sacrifice of the Rook. White should play instead 6.Kf5, when the best continuation (although one of many) for both sides is 6...Rd5+ 7.Kf6 Rd6+ 8.Kf7 Rd7+ 9.Ke6 Rd4 10.Ke5 Rb4 11.Rc8 Ra4 12.Rc6 Kg7 13.Kd5 Ra5+ 14.Kd6 Kg6 15.Kc7+ Kxg5 (sacrificing the g-Pawn) 16.Kb6 Ra4 17.a7, promoting the a-Pawn. The White King has just the right number of files between the Pawns to harrass the Black Rook and watch the g-Pawn.

I'm not sure what idea Levenfish & Smyslov missed in that last line. After 1...Rc4, they warned that White should avoid 2.a5 and play 2.Kh4 instead, but both moves are good for a win.