Uncountable FFKN rank for separable Banach spaces

Determine whether there exists a separable Banach space whose underlying metric-space structure, or whose structure expanded by its algebraic operations, has FFKN-Scott rank d714.

Background

The paper notes that separable Banach spaces can have highly complicated structures and asks whether the uncountable FFKN rank phenomenon can occur within this familiar class of metric structures. The question may be interpreted either for the metric language alone or for a language including the Banach-space operations.

References

Problem \arabic{problemcounter.} Is there a separable Banach space which has FFKN-Scott rank $\omega_1$?

— The rank problem for metric spaces  (2610.22351 - Tornquist, 15 Oct 2026) in Section “Comments and problems,” subsection “FFKN rank of Banach spaces and C$^*$-algebras”