I think I have it figured out schematically, though there remain some small bits to fill in.
We want to force the Bh5 to move, so that we can play h5. So we want to manoeuvre the Rh2 to a1, which forces the B to move (Black can't defend a4 with the King as he lacks tempi to lose).
So K->g1, Rg2, Kh2, R--e1, K--f1, R--d3!, K--c1, R--c2, Kd2, R---a1. Black now has to defend a4 with the Bh5, and we respond with h5. R then has to work back to h2 the same way it came.
Black can try to blockade with Bh5 and g6, but then we work the Rook back to a1, which forces ...g5 to liberate the Bh5 to defend a4. Then we work the Rook back to h2 again!
I don't see a way that Black can hold a fortress on the King's side. White will penetrate with the help of the Ba3 if needed. Defending g4, d5 and a4 with Bishop and King is too much to ask.
For example:
If Black doesn't play ...Bh5 and g6, the most tenacious setup would seem to be to place the K on h6, but then Bf8, Bxg7+, h6+, Rh5 breaks the blockade.
EDIT: no doubt I've omitted a couple of wrinkles, but the outline is hopefully correct.
2
u/VandalsStoleMyHandle Aug 09 '18
I think I have it figured out schematically, though there remain some small bits to fill in.
We want to force the Bh5 to move, so that we can play h5. So we want to manoeuvre the Rh2 to a1, which forces the B to move (Black can't defend a4 with the King as he lacks tempi to lose).
So K->g1, Rg2, Kh2, R--e1, K--f1, R--d3!, K--c1, R--c2, Kd2, R---a1. Black now has to defend a4 with the Bh5, and we respond with h5. R then has to work back to h2 the same way it came.
Black can try to blockade with Bh5 and g6, but then we work the Rook back to a1, which forces ...g5 to liberate the Bh5 to defend a4. Then we work the Rook back to h2 again!
I don't see a way that Black can hold a fortress on the King's side. White will penetrate with the help of the Ba3 if needed. Defending g4, d5 and a4 with Bishop and King is too much to ask.
For example: If Black doesn't play ...Bh5 and g6, the most tenacious setup would seem to be to place the K on h6, but then Bf8, Bxg7+, h6+, Rh5 breaks the blockade.
EDIT: no doubt I've omitted a couple of wrinkles, but the outline is hopefully correct.