Hello and thank you for the invite to your club 
I would say that after the exchanges along the d file, the move is Kg4. Once the rooks come off the board, I believe the position is simplified enough to the point where the kings need to centralize and control more of the board. g3-g4 also works but takes away the square from the king if he ever desires to get there.
The Process of Decision Making in Chess Volume 2: Practice positions and solutions. Position 10.2
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I don't agree with YungSeaTurtle.
First: in the question it was suggested, that the move was made before the exchanges. This point was missed by YST, but that may not be the main point.
After Kg4 black can easily play Kg6 and after that h5 can not be prevented. The white king must then retreat to a white field, whereupon black can play g4 with check. The result is, that 2 black pawns have moved to white fields, where the idea was to restrict them to black fields to limit the black bishop in its movements. g4 serves the purpoe better, but allows black to play a4 and similarly block the white a and b pawns
I would prefer a4. a possible black move h7-h5 can then still be met with g4 to fix the black pawns on the kingside. But I must admit that after f6-f5 instead of h5, g4 iis met by f4 and after the forced exchanges white's bishop is awquardly placed.
Summing up I think g4 must be the best move.
Ah yes, I totally misread and miscalculated haha. Good points you made. After reading back, the move were are discussing would come before the exchanges. Kg6 is a really good response to Kg4 but after g3-g4 what is stopping black from playing Kg6 anyways? Can he still push f6-f5 and try to create isolated pawns for white after fxg4?
1. g4 appears better locking up the kingside & then the white king could get to the left side of board to attack pawns eventually or kg4 kg6 then rook to d7, RxR, RxR. Then R-d7 attacking bishop then R-d8 wanting to trade but then Rook to B7 attacking pawn ?? The Rook then has to go to E6 to protect pawn & then white will gain an advantage on the right side of the board? Just a guess but trying to look ahead 4 to 5 moves....anybody agree or see that? I also notice p-g4 & then RxR so my 2nd plan wouldn't work....Thankyou....
Hello Mr. Spochman.....Hope your day is going well? Do you ever give out the answers to these challenges? Just curious because I always look back but never see exactly how the answer should be. Thank you.....Trace
Hi @Theprotege77
In case no one is on the right track, I try to offer some guidance, but not a conclusive solution, in order to allow new students in the future to enjoy the challenge as well.
Students with concrete questions raise them in our online/group lessons, where we have a great opportunity to discuss the theoretical background that helps approaching these challenges correctly.
Hello...okay gotcha & I understand. Thankyou for your time & feedback info! Trace
Solve the challenge below:
Diagram 10.2- how can white fight for an advantage in the given position? One of the ideas he may employ is to prepare the position for mass exchanges on the d file. So, how can white extract the most out of these exchanges? The arising position will be a same color bishops ending. Therefore, we will try to restrict the opponent’s bishop and create targets for our bishop, creating the conditions for our bishop to shine at the expense of the opponent’s piece.
We would like therefore to keep black’s pawns on dark squares, and this can be done in a number of ways- a4, c4, or even Kg4 to be played first before going for the exchanges. This way we try to maximize our winning chances.
Challenge: if white plays g4, clearly this fixes black’s pawns on dark squares on the kingside. Is this idea preferable to playing Kg4?