The attacking king must defend its own pawn while keeping the enemy king from reaching the opposition in front of the e-pawn; the only route uses the first rank.
One of Grigoriev's deepest king-and-pawn studies (1936). White has pawns on e2 and h4 against Black's h5; the white king is on h3, Black's on d5. The h-pawns are blocked, so White's winning chances rest on the e-pawn, but the black king is already in the centre.