End Game Tactics - Promoting Connected Pawns

A fairly even game progressed with Black maintaining an additional pawn gained from an early trade. This was carried into the end game but it was caveated with being a doubled pawn. White offered the exchange of the Rooks with 30. Re6+??. White's idea was that his King should be able to take the pawn chain starting on h6. What idea had he overlooked that Black could exploit to win the game?