Dehn invariant sufficiency conjecture for hyperbolic and spherical 3-space

Let $\operatorname{\mathcal P}(\mathbb X)$ be the scissors congruence group of polytopes in a 3-dimensional geometry $\mathbb X$, and let

$$\operatorname{\mathcal D}(\mathbb X):=\operatorname{Ker}\bigl(\delta\colon\operatorname{\mathcal P}(\mathbb X)\to\mathbb R\otimes\mathbb R/\pi\mathbb Q\bigr)$$

be the kernel of the Dehn invariant. Dehn invariant sufficiency conjecture. The volume maps

$$\operatorname{vol}\colon\operatorname{\mathcal D}(\mathbb H^3)\to\mathbb R$$

and

$$\operatorname{vol}\colon\operatorname{\mathcal D}(\mathbb S^3)\to\mathbb R$$

are injective. This is known in Euclidean 3-space by Sydler's theorem, while the corresponding hyperbolic and spherical cases remain open; the conjecture would imply that both Dehn-invariant kernels are countable.