My gut feeling is that white should try to exchange h3 for h7 to get relief from defending this pawn. After 1. h4 Bg4 (presuming black tries to prevent it) 2. Nd1 Bf4 3. Ne3, black faces the choice between burying his bishop on h5, exchanging bishop for knight (not advised) and allowing h5 to go through. After the pawn exchange, white has his hands (more or less) free to start a line opening action on the queen side to get his rook working. Of course he must now keep the passed pawn g6 in check, but black will hesitate to push it and deliver f5 to the white knight.
Just a few thoughts but certainly not exhaustive. I am not strong enough as a chess player to properly judge this position though I am quite a bit better than my blitz rating suggests. Hopefully, you'll get an IM to look at it and give his views.
I am right now doing some home analysis (which perhaps may be flawed and worthy of a patzer) on the Bronstein Queen sac line, and after move 30 or so I reached this position: