Degree-4 Sum-of-Squares certificate for nonexistence of 7 MUBs in C^6

Develop a degree-4 Sum-of-Squares proof that there do not exist seven mutually unbiased bases in C^6.

Background

Sum-of-Squares relaxations provide a proof system to certify polynomial constraints. Prior work shows certain degree-2 relaxations cannot rule out seven MUBs in C6; it remains unclear whether higher-degree SoS can.

A degree-4 SoS certificate would make significant progress on MUB(6) and illuminate the power and limitations of low-degree SoS for nonconvex algebraic feasibility problems.

References

It is however unclear whether higher degree levels of Sum-of-Squares can provide such a proof. Is there a Sum-of-Squares degree 4 proof that there are no $7$ Mutually Unbiased Bases in $\mathbb{C}6$?

— Randomstrasse101: Open Problems of 2025  (2603.29571 - Bandeira et al., 31 Mar 2026) in Open Problem, Section “Mutually Unbiased Bases, ETFs, and Zauner’s Conjecture (ASB)” (Entry 11)