Neumann's Bloch-group generation conjecture for hyperbolic 3-manifolds

Let $F \subset \mathbb{C}$ be a number field not contained in $\mathbb{R}$, and let $\mathcal{M}$ be the set of manifolds with invariant trace fields contained in $F$,

$$\mathcal{M}:= \left\{ M \cong \mathbb{H}^3/\Gamma \mid \mathbb{Q}(\operatorname{tr} \Gamma^{(2)}) \subseteq F \right\}.$$

Let $\mathcal{N} = \left\{[M] \mid M \in \mathcal{M}\right\}$ be elements of the Bloch group determined by manifolds in $\mathcal{M}$. Neumann's conjecture. Integral combinations of elements of $\mathcal{N}$ generate $\mathcal{B}(F)$. This conjecture predicts that hyperbolic 3-manifolds with invariant trace fields inside $F$ account for the entire Bloch group of $F$; the paper presents experimental evidence but does not establish the claim.