Near-perfect discrepancy for connecting-line boards

Prove or disprove the conjecture that, for every noncollinear n-point set P in the plane, the maximum signed discrepancy of the board whose switches are all connecting lines satisfies F(P)=n-o(n).

Background

The main theorem establishes the universal lower bound F(P) \geq n/3. The conjecture proposes a substantially stronger asymptotic result: regardless of the geometry of the noncollinear point set, the guaranteed maximum discrepancy should approach n up to a sublinear error term.

Because F(P) is defined as the worst-case, over initial weight assignments, of the best discrepancy attainable through connecting-line switches, the conjecture asserts that nearly all lights can always be turned on asymptotically.