Minimum achievable absolute discrepancy in the geometric variant

Determine the minimum absolute value of the discrepancy that can be achieved, for every initial weight assignment, in the purely geometric variant of the Gale–Berlekamp switching game whose switches are the connecting lines of a noncollinear point set in the plane.

Background

The paper briefly considers a different objective from maximizing signed discrepancy: minimizing the absolute discrepancy, thereby balancing the numbers of positive and negative weights as closely as possible. For the classical square-board Gale–Berlekamp game, the corresponding question is resolved by results of Komlós and Sulyok and of Beck and Spencer.

The analogous quantity for the purely geometric variant—where the board is a noncollinear planar point set and each connecting line is a switch—is left as an explicit question.