Nahm’s conjecture (Bloch-group vanishing implies modularity)

Establish that for any positive-definite symmetric matrix A, if for every solution X=(X_1,…,X_r)∈(0,1)^r of the Nahm equation 1−X_i=\prod_{j=1}^r X_j^{a_{ij}} (i=1,…,r) the associated Bloch-group class [X]∈B(ℂ) vanishes (condition (a)), then there exist B∈ℚ^r and C∈ℚ such that the Nahm sum f_{A,B,C}(τ) is a modular function (condition (c)).

Background

The authors recall the classical formulation: condition (a) is Bloch-group vanishing for all solutions of the Nahm equation; (b) is vanishing for the distinguished solution in (0,1)^r; (c) is the existence of B and C making the Nahm sum modular. Known implications include (c)⇒(b) (Calegari–Garoufalidis–Zagier), and counterexamples show (b)↛(a),(c) and (c)↛(a).

The belief that (a)⇒(c) captures Nahm’s conjecture, which has only been proved in rank one. Proving this would also imply the ADE/T folklore conjecture via Lee’s result on the Bloch group for A(X,Y).