There is a ton of mathematical literature about “tiling / tessellation”, but perhaps this particular type of problem has not yet been explored in the literature.

I think the chains and their intended liberties can be viewed as tiles, but the liberties are allowed to overlap when tessellating. These problems could be stated as: how to tessellate the 5x5 grid with a set of tiles (where each tile is a possible chain plus its intended liberties), but the tiles are allowed to partially overlap (on the liberties)? Specifically, the original post asks for tessellations that use the maximum number of tiles.

However, there is a complicating wrinkle to this problem: the points adjacent to the chains that should not be liberties must be filled with stones of a different color or the edge of the board. With only two colors to work with, this adds a complication that potentially reduces the possible solutions.

Thus, I conjecture that allowing more than two colors of stones would increase the solution space. I wonder if “four colors suffice” for achieving the maximum, or if a different number is the most one would need to maximize their score.

What are your best solutions with more than two colors of stones?

Now, I think the problem has at least three parameters to consider: size of the board, number of stone colors, and number of liberties per chain. Of course, other generalizations, like rectangular boards, wider bounds on the liberty count (instead of a single fixed number), etc., are also possible.

@Vsotvep built a tool for playing Diplomatic Go that could be useful here:

https://vsotvep.github.io/MulticolorGo.html?st=&bs=5