Existence of guarded Fraïssé spaces that are not cofinally Fraïssé

Determine whether there exists a separable Banach space that is guarded Fraïssé but not cofinally Fraïssé, or prove that no such space exists.

Background

Guarded Fraïssé is a weakening of weak Fraïssé tailored to yield a metric Fraïssé correspondence via prehomogeneity-type conditions. Cofinally Fraïssé is an intermediate notion between guarded and weak Fraïssé that is known to imply guarded Fraïssé. While separations are known in the field of countable first-order structures, it is unknown in Banach space settings whether guarded Fraïssé can fail to be cofinally Fraïssé.

Clarifying this would refine the landscape of approximate homogeneity notions for Banach spaces and their interrelations.

References

It is actually not known whether there exist guarded Fraïssé Banach spaces that are not cofinally Fraïssé (although such examples exist in the setting of countable first-order structures, see ).

— Isometric rigidity and Fraïssé properties of Orlicz sequence spaces  (2604.02080 - Rancourt et al., 2 Apr 2026) in Section 4 (Cofinally Fraïssé Orlicz sequence spaces)