Contractibility of the moduli space in the Gromov–Hausdorff topology

Determine whether the moduli space of locally conformally flat metrics with positive scalar curvature on a smooth manifold homeomorphic to a spherical space form but not diffeomorphic to the sphere is contractible in the Gromov–Hausdorff topology.

Background

The paper proves that, for higher-dimensional smooth manifolds homeomorphic to spherical space forms but not diffeomorphic to the sphere, the moduli space of locally conformally flat metrics with positive scalar curvature is either empty or contractible in the quotient topology induced by the smooth topology on metrics. The author also states that conformal classes are dense in the moduli space for the Gromov–Hausdorff topology.

Contractibility in the Gromov–Hausdorff topology is not established by the deformation-retraction argument used for the smooth-topology moduli space, and is explicitly left unresolved.