Exact nonlocality threshold of two-qubit Werner states
In plain words
A Werner state mixes a perfectly entangled pair with random noise. Below some noise level all projective-measurement results can be reproduced by classical hidden variables, and that exact level is unknown.
Precise statement
For $\rho_W(v) = v \left|\psi-\right\rangle\left\langle\psi-\right| + (1 - v) I/4$ ($I = 4x4$ identity matrix) with local projective measurements, the critical visibility above which some Bell inequality is violated is $v_c = 1/K_G(3)$, where $K_G(3)$ is the order-3 real Grothendieck constant. Current bounds $1.43670 \le K_G(3) \le 1.4546$ give $0.6875 \le v_c \le 0.69604$. Determine $v_c$.
What would settle it
Matching local-model and Bell-inequality constructions fixing $v_{c}$ to all digits, or a closed form for $K_{G}(3)$.
Status in the literature
Unverified note
Lower bound $K_{G}(3) \ge 1.43670$ (Designolle, Vertesi, Pokutta, PRA 2026, arXiv:2409.03739) and upper bound $K_{G}(3) \le 1.4546$ from Frank-Wolfe local models (Designolle et al., 2023) give $0.6875 \le v_{c} \le 0.69604$.