Nahm's conjecture on modular triples and Bloch-group torsion

Let $A=(a_{ij})$ be a positive definite symmetric $r\times r$ matrix with rational entries. Consider the system

$$x_i=\prod_{j=1}^{r}(1-x_j)^{a_{ij}},\qquad i=1,\ldots,r.$$

For a solution $\mathbf{x}=(x_1,\ldots,x_r)$ in a number field $F$, let $\xi_{\mathbf{x}}=[x_1]+\cdots+[x_r]$ be the associated element of the Bloch group $\mathcal{B}(F)$. A triple $(A,B,C)$ is called a modular triple if $B$ is a rational vector of length $r$, $C$ is rational, and

$$f_{A,B,C}(z)=\sum_{n\in(\mathbb{Z}_{\geq 0})^r}\frac{q^{\frac12 n^tAn+B^tn+C}}{(q)_{n_1}\cdots(q)_{n_r}},\qquad q=e^{2\pi i z},$$

is a modular function, where $(q)_n=(1-q)\cdots(1-q^n)$. Nahm's conjecture. The following are equivalent: (i) for any solution $\mathbf{x}$ of the system, the element $\xi_{\mathbf{x}}\in\mathcal{B}(F)$ is a torsion element of $\mathcal{B}(F)$; (ii) there exists a modular triple $(A,B,C)$.