Plasticity of unit balls of Banach spaces (single-space and two-space versions)

Determine whether, for arbitrary Banach spaces X and Y (including the special case X = Y), every non-expansive bijection between the unit balls B_X and B_Y is an isometry; equivalently, establish whether the unit ball of an arbitrary Banach space is plastic and whether this property extends to pairs of (possibly different) Banach spaces.

Background

A substantial body of work studies when non-expansive bijections between unit balls of Banach spaces must be isometries, with many positive results in particular settings and under additional assumptions. Nonetheless, a general resolution for arbitrary Banach spaces remains elusive.

The authors highlight that, despite progress (including conditional results and examples in special classes), there are no known counterexamples of Banach spaces with non-plastic unit balls. The question persists both for a single space (self-bijections of its unit ball) and for bijections between unit balls of two possibly different Banach spaces.