Here is a theoretical ruleset for the proposed variant of chess:

1. Expand the standard chessboard to include gray squares.

2. Allow bishops to move into and out of gray squares freely, like the king piece, but forbid capturing pieces on gray squares.

3. Follow the standard chess rules for all other pieces with respect to black and white squares.

4. Continue the rules for pawn movement, king checkmate, and stalemate from standard chess.

5. Kings will still be subject to check and checkmate, and all other rules pertaining to these conditions will remain in effect.

6. Stalemate rules will be the same as standard chess.

This ruleset aims to make it easy to transition from standard chess to this variant while introducing the new mechanics. The addition of the gray squares and the ability of bishops to move freely on them will lead to more careful positioning and add more depth to the decision-making process. The ability to change the bishops' color affinity may add some strategic flexibility, but it should be carefully balanced to prevent either side from gaining an unfair advantage.
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