Nahm's equations and modular forms

We consider certain $q$-series depending on parameters $(A,B,C)$, where $A$ is a positive definite $r \times r$ matrix, $B$ is an $r$-vector and $C$ is a scalar, and ask when these $q$-series are modular forms. Werner Nahm has formulated a partial answer to this question: he conjectured a criterion for which $A$'s can occur, in terms of torsion in the Bloch group. The conjecture was proved by Don Zagier for $r=1$. Recently me an Sander Zwegers found several counterexamples for $r\ge 2$, so the correct formulation of Nahm's conjecture became an interesting open question.