Hilbert-space classification of the Gromov–Hausdorff–Prokhorov space

Determine whether the Gromov–Hausdorff–Prokhorov space of isomorphism classes of nonempty compact metric spaces equipped with arbitrary Borel probability measures, without a full-support assumption, is homeomorphic to the real separable infinite-dimensional Hilbert space.

Background

Earlier in the paper, the unmeasured Gromov–Hausdorff space is embedded as a retract of a Gromov–Hausdorff–Prokhorov space of measured compact metric spaces. The construction allows measures that are invariant and have full support, while the question concerns the larger space with arbitrary Borel probability measures.

The paper asks whether this larger measured space has the same Hilbert-space homeomorphism type as the unmeasured Gromov–Hausdorff space.