Wrong assessment in Fundamental Chess Endgame by GM Karsten Müller, the endgame Guru

The book gives 1.c5+- but I claim Black can hold his stand. Draw with ...b5!

@notmtwain: I think @BL4D3RUNN3R was saying 1. c5 b5 is a draw, but some supporting analysis could be helpful.

So actually I got some analysis but I would like to hear some further thoughts.

My first thought was white plays Kh4 and then advances the kingside pawns and will be winning, but you can play ...g6 and meet g4 with ...g5 and it seems white can’t progress that way after all. So it does look like Black might be holding. 

In any case, I would play ...b5 without much thought, since everything else clearly loses easily. 

Fun fact: the tome is written by Mueller & Lamprecht and this game was played between them both.

So it is fair to say that it is probably not won but a draw. My analysis:

I can't find any win either. g4 is answered with g5 and h4 with h5. My best try transposed in exactly the same queen+ pawn vs queen endgame which looks like a draw.