Smooth classification of the parameter space in the sphere case

Establish whether the parameter space of an unoriented transitive family of closed embedded hypersurfaces on a manifold diffeomorphic to the sphere is diffeomorphic to real projective space.

Background

Theorem B proves that when the ambient manifold is diffeomorphic to the sphere, the parameter space of the transitive family is homeomorphic to real projective space, while its universal cover carries an oriented transitive family of embedded spheres. The paper does not obtain a diffeomorphism classification of the parameter space in this case and explicitly records the stronger smooth conclusion as unresolved.