Convergence of the universal sum-of-squares hierarchy

Prove that the universal sum-of-squares hierarchy for generalised incompatibility robustness satisfies \lim_{t\to\infty}\chi^g_{k,t}(n)=\lambda_{k,n}/k for every k,n\geq2, and consequently establish \chi^g_3(n)=\lambda_{3,n}/3 whenever a 3-fold unbiased measurement with n outcomes exists, including maximal incompatibility of the Hadamard--Clifford triples.

Background

The paper develops a symmetry-reduced sum-of-squares hierarchy that supplies universal lower bounds on the generalised incompatibility robustness. Numerical computations agree to high precision with \lambda_{3,4}/3 for the constructed four-outcome 3-fold unbiased measurements, and broader numerical data support the same value for additional parameter pairs. An analytical convergence proof is not obtained; the conjecture would convert these numerical observations into a universal incompatibility theorem, even when no finite-dimensional k-fold unbiased measurement is known.