I haven’t tried your problem, but I noticed once obvious thing:
Using Symmetric is a hint to tell JuMP that the matrix GG is symmetric. If the matrix is not symmetric, then you can get incorrect results because JuMP will not add constraints to ensure the matrix is symmetric.
Try removing Symmetric: @constraint(model, GG >= 0, PSDCone()).
From a conceptual point of view though, I think you need to reconsider what you’re trying to do. The solver will find you a solution that satisfies the constraints as you have written them. If you want a different solution, it means that you are missing some constraints. That might, for example, mean adding a constraint like @constraint(model, G[1, 1] == 1) to fix one of the variables to a known value, around which the rest of the solution can be built.