Equivalence of unique and strongly unique preduals

Determine whether the properties of having a unique predual and having a strongly unique predual are equivalent for dual Banach spaces.

Background

The paper distinguishes two notions for a dual Banach space: a unique predual means that all preduals are linearly isometric, whereas a strongly unique predual means that every surjective linear isometry onto a dual Banach space is weak*-to-weak* continuous, equivalently that the canonical dual space has exactly one predual subspace under the standard identification. The authors note that strong uniqueness formally implies uniqueness, but whether the converse holds is unresolved.