Hilbert-cube type of the Nakajima–Shioya compactification

Determine whether the compactification of the Gromov–Hausdorff space constructed by Nakajima and Shioya is homeomorphic to the Hilbert cube.

Background

Nakajima and Shioya construct a compact metrizable compactification containing a dense topological copy of the unmeasured Gromov–Hausdorff space. The paper identifies the homeomorphism type of the dense space but leaves the topological classification of this compactification unresolved.