Hilbert-space classification of the unit-diameter subspace

Determine whether the subspace of compact metric spaces of diameter one, equipped with the Gromov–Hausdorff topology, is homeomorphic to the real separable infinite-dimensional Hilbert space.

Background

The full Gromov–Hausdorff space is shown to be homeomorphic to Hilbert space. The paper asks whether the normalization obtained by restricting to compact metric spaces of diameter exactly one preserves that homeomorphism type.