  Nahm's conjecture relates $q$-hypergeometric modular functions to torsion
elements in the Bloch group. An interesting class of such functions can be
(conjecturally) obtained from a pair $(X,X')$ of diagrams, each of which is
either a Dynkin diagram of type $ADE$ or a diagram of type $T$. Using
properties of Y-systems, we prove that for a matrix of the form
$A=\mathcal{C}(X)\otimes \mathcal{C}(X')^{-1}$ where $\mathcal{C}(X)$ and
$\mathcal{C}(X')$ are the corresponding Cartan matrices, every solution of the
equation $\mathbf{x}=(1-\mathbf{x})^A$ gives rise to a torsion element of the
Bloch group.
