Optimal 2×2×3 nonlocality-strength conjecture

A $2\times2\times3$ nonlocality proof is a Bell experiment with two settings for each party and three outcomes for the relevant measurement type. Its statistical strength is the information-theoretic strength used in the paper to quantify evidence against local-realist theories. The displayed bipartite state is written in the computational basis $\lvert 1,1\rangle$, $\lvert 2,2\rangle$, and $\lvert 3,3\rangle$.

Optimal 2×2×3 strength conjecture. The strongest possible $2\times2\times3$ nonlocality proof has statistical strength $0.077$, and it uses the bipartite state

$$0.6475\lvert 1,1\rangle+0.6475\lvert 2,2\rangle+0.4019\lvert 3,3\rangle.$$

The paper contrasts this proposed optimum with the maximally entangled state, whose reported statistical strength is only $0.058$, and notes the resulting distinction between entanglement and statistical nonlocality. The claim is based on preliminary numerical investigations and remains open.