Is it possibile to break through this fortress?

Doesn't seem too hard. How is white supposed to maintain his formation? Only safe moves to make are with the h pawn. Black brings the king down through the h-file, grabs the pawn and white has no moves, no?

Now that I've tried it, I can't do it, so disregard what I said, lol.

Stockfish gives black the advantage, but he doesn't recognise fortresses usually

King gets to a3, via d7, c6, c5 and b4. And the result gets 0-1.

I don't think it is possible. From what I have tried, you might have some chances if you have a pawn on a6, then there are some queen sac ideas on a7 if the king and the bishop are too far away. With the pawn on the position given, I managed to make it passed but the counterplay on the other side is too fast. If you guys want to try this position, I recommend trying with the pawn on a6 first and see if you can win, then proceed with the "original" given position.

If Kb1 then Qe1+ and when the king moves back to b2, h5 and the king will have to move, then the bishop. Then white's position falls.

I did a different variation, here it is: (PGN)

[Variant "From Position"]
[FEN "8/7p/4k3/5p2/2P1pB2/1P2P3/1KBq3P/8 w - - 1 1"]

1. h4 Qb4 2. h5 Qe1 3. h6 Qd2 4. Bg5 Qg2
5. Bf4 Qd2 6. Bg5 Kd7 7. Bf4 Ke7 8. Kb1 Ke6
9. Bg5 Qc3 10. Kc1 Kf7 11. Bf4 Kg6 12. Kb1 Kh5
13. Kc1 Kg4 14. Kb1 Kf3 15. Kc1 Qa1+

After this Black can win.

#5
Indeed Bf4 is overloaded; he cannot defend e3 h2, and the black squares a3-b2-c3 around the white king all at once.

This plan of bringing the king in looks promising but can someone show the winning line? The pawn will never be on h2, this is the key because the queen can't exert more pressure than now with the pawn on h2. As soon as the king is on c5, the bishop should be on b4 or a3 and check the king away with tempo and then come back to its defensive post.

The #7 line,10.Bd1 and the king can't penetrate.

But interesting ideas guys, I want to break this fortress as well :)

Turns out both the plans that were mentioned before works. The key is to bring the king. Either from the queenside or the kingside, with the support of the queen or zugzwang, I am convinced now.

1.h4 Qb4 2.h5 (2.Bg5 Qe1 3.Bb1 Kd6 4.Bc2 Qd2 5.Kb1 Kc5 6.Be7+ Kc6 7.Bg5 Qe1+ 8.Kb2 h6 9.Bxh6 Qxh4 10.Bf4 Qh8+ 11.Kb1 Kc5) Qe1 3.h6 (3.Bg5 Qe2 4.h6 Qd2 5.Bf4 Kd7 6.Bg5 Kc6 7.Kb1 Qg2 8.Bf4 Qd2 9.Bg5 Qd6 10.Kc1 Kc5 11.Bf4 Qf6) Qd2 4.Bg5 Qg2 5.Bf4 Qd2 6.Bg5 Kd7 7.Bf4 Ke7 8.Kb1 Ke6 9.Bg5 Qc3 10.Bd1 Kd6 11.Bc2 Ke5 12.Bd1 Kd6 13.Bc2 Kf7 14.Bd1 Kg6 15.Bf4 Kf6 16.Bc2 Kf7 17.Bd1 Kg6