Complicated position


Immediate 1. h6 doesn't seem to work, because of how active black rooks can become through certain forced lines. For example:
- and the black has powerful connected rooks and, at least, has an easy draw if the white rook moves out of ABCDE lines.
I think 1. Rg2 is promising. If black decides to exchange one pair of rooks, then white doesn't let the black pawns move due to the Rb7+ and Rxb5/Rxb4, and his king can try to move towards the h pawn to support it. Otherwise, white can get "pigs" on the 7th line, and something can happen there. In case of exchange, a sample line:
- and the position is pretty sharp, with both players having chances to promote a pawn before the opponent.

Immediate 1. h6 doesn't seem to work, because of how active black rooks can become through certain forced lines. For example:
Yes, good one. That preliminary ...Rh4+ to get White's king in the corner first, followed by ...Rhg4 to threaten mate at g1 is nasty. I don't see that White can get out of that bind.
A couple of days ago I started playing 3 min games after a looooooong period of only bullet games. One of the games got to this position:
In the real game, my time was running out, I panicked and blundered: 36.Rxc6?? Rh4#. However, I find the position very interesting to analyze.
Black has an extra pawn and seems to be winning one more pawn (a6), but white has a very strong supported pawn at h5. The engine says that after 36. Rb7+ Kc8 37. Rxc6+ Kd8 38. Rb8+ Kd7 the position is dead even. However, it does not recommend a more intuitive line for black 36. Rb7+ Ka8 37. h6 Ra1+ (37...Rxa6? 38. h7 Rh8 39. Rg7, winning a rook) 38. Kh2 Rh8 39. h7, with an incredible advantage for white.
Other than the 36. Rb7+ lines, are there any ways for white to attempt to gain some advantage in this position?