Or a perpetual check on e1 and e4 squares if Qs not swapped?

After 40.Qxe7 Qe8>

Since no experts have taken this on, I'll try it.

Fir tge saje if argument, we'll say the game proceeds with <37...Ka6 38.Qd6+ Kb7 39.Qxe7 Qe8>. (There may be better moves, but I don't think they affect the basic ideas that much.)

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Having bishops of opposite-colors can affect play in several ways. In positions where there is a lot of material on the board, the side with the initiative often has an extra piece, for practical purposes, as the opposite-bishop cannot oppose its attacking counterpart.

As the amount of material decreases, the opposite-bishop becomes more of a factor as it may be able to set up a blockade which the other side does not have enough force to break.

Now there is not a lot of material in this position, but there are queens, so White consider continuing his attack or at least the pressure. The idea might be something like 40.Qb4+ and 41.Be5. Now Blacks' queen can't get into play quickly, and in any case doesn't dare stray far from the defense since the light-squared bishop can't defend a dark-squared attack. White will begin pushing his passed pawns, and the pressure will eventually become too much.

Keeping queens would probably be good for White, but the pure bishop ending would be good as well because there are so many pawns left, scattered all over the board. Here are a couple of schematic position:

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This is a trivial draw, showing both the blockade and the draw by insufficient material. . As long as black's bishop remains mobile, his king just sits on c7 and refuses to move out of the way. If the pawn ever does move to c5, Black just snaps it off.

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Drawn again. If White plays 1.c5+, Black snaps off both pawns. If he plays 1.b5, B lack need only control c5 and White does not have enough force to break the blockade. If the White king guards b3, it can never support the c4-c5 push as well. Not as trivial as the one-pawn position, but Black holds.

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Now life is getting difficult. Since White has two pawns supporting the crucial c4-c5 push, breaking the blockade. There are trick positions where Black can draw against three pawns, but they are rare: White, with careful play, can fully expect to win.

Now, White doesn't have three connected pawns here, but he has something just as good: separated pawns, and pawns on both sides of the board. Let's trade the queens, play b2-b3 and c3-c4, and say Black trades on c4. (There are probably better moves, but this is the general plan.):

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If the kingside pawns were off the board Black might have a chance if he could give up his bishop for the c # d pawns, he could draw against the rook-pawn plus wrong-colored bishop. But those kingside pawns are there, so such a pawn can't work.

One more point: having separated pawns in opposite-colored bishop endings can be better than connected pawns. A blockade works best over a limited playing area, and the wider the area the harder it is for a blockade to work.