Almost made it...
I picked this up from the web with the question, can black draw? The webmaster of the Philippine Chess Chronicles, Francis Buenaventura, playing black, figured out what could be the best idea in this position, stalemate. So, he played the very clever 1...Rc4+. But, is it enough? Play continued with 2. Kd2 Rc2+ 3. Ke3 and the white king ultimately found shelter behind his C and D pawns. However, 2...Rc2+ was not what the doctor ordered in this position. That move let's the cat out of the bag.
Let's take the advice of Capablanca when he urged players to think in schemes in the endgame and not get tangled up in analysis.
First, let's recognize that the white king is completely exposed to checks below the fourth rank. Therefore, it is imperative that the king is not allowed to escape by giving up the demarcation line on the fourth rank created by the rook. The rook must corral the white king below the fourth rank or lower. You can already visualize the continuous checks by the rook along the fourth rank. If the rook takes the A pawn with a check and the king recaptures, it would be a stalemate. However, if the king does not recapture, black has just created a free square on b5, and the stalemate is no longer possible. The problem with black's idea is that there is a hole on h4 from which the rook cannot give a check. The queen controls that square. Because of this, black cannot draw this position. The white king heads for the h4 square to in order to cross the border.
Ok, now, let's see it in action with some simple analysis: 1... Rc4+ 2. Kd3 Rd4+ 3. Ke3 . Re4+ 4. Kf3 Rg4+ 5.Kh3 and there you have it. Black has a problem. If black plays 5...Rg3+ , then 6. Kh4 Rg4+ 7. Kh5 and there is no check on g5 (queen). Black has to move, and then white gets 8. Qc8+ in, game over.
One final note, I think if the rook was on the third rank and the king on the second, then black's perpetual checks will draw. There, I hope that I have not strayed too far from Capablanca's dictum.