The position very much looks to be a draw. The white king can't advance easily as the black rook stays on the third rank. All the maneuvers to bring back the white rook to help advance don't seem to work as both f4 and d4 can be weak. Furthermore the black king can stay on the d file to cover the e7 and c7 square while always having Kd7 if White attacks the weak e6 or c6 pawn from the side.
Therefore the only way to make progress for White is to play 0...Kd8 1.f5 exf5 2.Rxf5
Now White faces a similar problem. The d4 pawn is weak so as soon as White blocks with the rook on the third rank the black rook will move to the fourth rank and keep the d4 pawn attacked. For instance 2...Rh3 3.Rf3 Rh4 . To make progress White would have to block the fourth rank. However if White plays Ke3 Black can respond with Re4+ forcing the king to d3 from which it can not support a move like Rf4 blocking the fourth rank.
This of course is not a proof but I am very sure that for the reasons outlined here, the position is a draw under all circumstances.