Determine the optimal zero-key attack threshold for the Magic Square game

Determine whether the winning probability approximately 0.935 achieved by the described adversarial strategy for the Magic Square game is optimal, and characterize the largest winning probability compatible with Eve learning all key bits.

Background

The appendix describes an attack in which the devices reveal the key outputs to Eve while achieving a Magic Square winning probability of approximately 0.935. This establishes that a winning score strictly above the classical value 8/9 does not by itself guarantee positive key entropy at that threshold.

The authors report an SDP upper bound of approximately 0.945, leaving a gap between the exhibited attack and the best known upper bound. Closing this gap would determine the exact threshold at which the Magic Square-based device-independent key protocol can guarantee nonzero secrecy.