39.Qxb3 Rf8+, 40.Ke2 Re8+, 41.Kf3

(saving us a tempo because that gives us a chance to get the rook to the 7th rank that's worrying us out of our minds!)

41...Re7, 42.Qd5

(I tried to select/find the White move that would most complicate things/make things difficult for us. But that's not to say they're going to play that move!)

42...Be5, 43.Qc5+

I VOTE FOR 40.Ke2 Re8+

TWO LINES TO CONSIDER:

Line #1) 39.Qxb3 Rf8+, 40.Ke2 Re8+, 41.Kf3 Re7, 42.Qd5 Be5, 43.Qc5+ Kd7

(Stalling. d6 & d5 are the best squares for our King)

44.Qb5+ Kd6, 45.g4

(Trying to divide & distract our forces by advancing on both sides of the board. But that's only a delusion!)

45...Rc7, 46.Kg2 Rf7, 47.a5 Rc7

(It looks like useless repetition. But,
the point is to stall, & allow the pieces to get to their most ideal squares. I'm using their pawn movements to do that.)

48.a6 Bd4, 49.Qd3 Kc5, 50.Qg6 Rc6, 51.Qh5+ Kb6, 52.h4 Rc5, 53.Qg6+ Bf6

(Note: King is close to a-pawn, Rook covers 5th rank & Bishop & h-pawn cover g5, in addition to the Rook!)

54.Kg3 Rc6, 55.Kh3

(Seemingly wasting time. But, if 55.Kf4 or 55.g5, Be5+, forking the Queen!)

55...Kxa6

Line #2) 39.Qxb3 Rf8+, 40.Ke2 Re8+, 41.Kf3 Re7, 42.Qd5 Be5, 43.Qc5+ Kd7, 44.Qb5+ Kd6, 45.h4

(Note: Unlike in previous line, this advance robs the King of the vital g2 square!)

45...Rf7+, 46.Kg4

(Other moves are no better & take the King far from the Kingside pawns, which can't possibly help them in any way, shape or form!)

46...Rf4+, 47.Kh3 h5, 48.g3 Rg4,

(I tried my best to get to e1 or d1, followed by h1 with mate. But moving the rook to either e4 or d4 resulted in a series of checks from White that only let our King & their Queen dance all over the centre!)

49.Qe8 g6,

(Notice that even though the pawns are no longer on dark squares, where the bishop & rook, & perhaps King, can co-operate to protect them, they're still protected in this formation!)

50.Qf8+ Kd5, 51.Qf2 Rxa4, 52.Qd2+ Ke6, 53.Qc2 Rg4, 54.Qf2 Rxg3+, 55.Qxg3 Bxg3, 56.Kxg3 Kf5

(Their h-pawn falls next & we have TWO pawns that can & will be queened!)

As for the game:

1) Kd3 abandons the Kingside pawns. I don't about you or the others, but I think that that's just plain fool-hardy. You have to give them more credit than considering they'd play something like this! They're not that novice players!

2)41...Rf8+ wastes time. Only the bishop or rook can protec the g7-pawn. But, remember the a-pawn as well. Only the rook can cover the entire 7th rank! Also, on the 7th rank, it's closer to BOTH bishop & King, to assist or be protected by them. You &/or may/will disagree with me. But that's what I think.