Remaining quartic case with sqrt(2) in K

Determine whether any quartic totally real number field K with sqrt(2) ∈ K admits a positive definite classical ternary quadratic form over OK that is universal; equivalently, resolve the remaining quartic-case obstruction in the classification of fields admitting universal ternary classical forms.

Background

The paper proves strong constraints implying that no quartic field K can admit a universal ternary classical form under either the condition sqrt(2) ∉ K or the presence of a nonsquare totally positive unit. This leaves a single quartic scenario unresolved: the case where sqrt(2) lies in K and the unit conditions do not rule out universality.

The authors explicitly note this as the only open subcase in their quartic-field program, indicating a targeted direction to complete the quartic classification.