Separation of scissors congruence classes by volume and generalized Dehn invariants

Let polytopes be given in Euclidean, spherical, or hyperbolic geometry. Their volume and generalized Dehn invariants are defined by summing suitable measures over faces of each dimension, with the angle measure taken in the corresponding sphere.

Volume–Dehn separation conjecture. In Euclidean, spherical, and hyperbolic geometries, do the volume and generalized Dehn invariant separate the scissors congruence classes of polytopes?

A positive answer would provide a complete set of invariants for scissors congruence in these geometries, extending the Dehn–Sydler theorem in Euclidean dimensions three and four and the work of Dupont and Sah on three-dimensional spherical and hyperbolic spaces. The source poses this as a question and gives no resolution.