Box-topology metric-measure space classification

Determine whether the space of mm-isomorphism classes of metric measure spaces equipped with the box topology is homeomorphic to the real separable infinite-dimensional Hilbert space.

Background

The paper proves that the ordinary unmeasured Gromov–Hausdorff space is homeomorphic to the real separable infinite-dimensional Hilbert space. It then asks whether an analogous classification holds for metric measure spaces under the box topology.

The concentration-topology analogue is known to have a negative answer because that space is not Baire, whereas the Hilbert space is Baire. The box-topology case remains unresolved in the stated question.