CHSH optimality conjecture for correlated 2×2×2 Bell proofs

A $2\times2\times2$ proof is a Bell-nonlocality proof with two settings and two outcomes for each of two parties. A setting distribution may be correlated when the parties' measurement settings are sampled jointly rather than independently. CHSH denotes the Clauser–Horne–Shimony–Holt proof.

CHSH optimality conjecture. Among all $2\times2\times2$ proofs, and allowing correlated setting distributions, CHSH is best.

This conjecture concerns the maximal statistical strength among binary two-party Bell proofs when correlated sampling is allowed. The paper reports numerical analysis but does not establish the asserted optimality, so the conjecture remains open.