Neumann's volume realization conjecture for boundary-unipotent representations

Let $M$ be a $3$-manifold, let $n$ be a positive integer, and let $\rho$ be a boundary-unipotent representation of $\pi_1(M)$ in $\mathrm{SL}(n,\mathbb C)$ or $p\mathrm{SL}(n,\mathbb C)$. Write $\operatorname{Vol}_{\mathbb C}(\rho)$ for its complex volume, and let $\operatorname{Vol}_{\mathbb C}(M_i)$ and $\operatorname{Vol}(M_i)$ denote the complex and real volumes of a hyperbolic $3$-manifold $M_i$. Neumann's conjecture. There exist hyperbolic $3$-manifolds $M_1,\dots,M_k$ and integers $r_1,\dots,r_k$ such that

$$\operatorname{Vol}_{\mathbb C}(\rho)=\sum r_i\operatorname{Vol}_{\mathbb C}(M_i)\in \mathbb C/i\pi^2\mathbb Q.$$

In particular,

$$\operatorname{Vol}(\rho)=\sum r_i\operatorname{Vol}(M_i)\in\mathcal R.$$

This is presented as a consequence of Walter Neumann's stronger conjecture that the Bloch group of a concrete number field is generated modulo torsion by hyperbolic manifolds with invariant trace field contained in that field. The status of the stated volume consequence is not resolved in the supplied text.