Usefulness of the constant-term representation (3.11) for proving modularity

Ascertain whether the identity X_r(1,1,...,1; q) = (q; q)_∞ ∑_{n_1,...,n_r ≥ 0} 1 / [(q; q)_{2 n_1} ... (q; q)_{2 n_r}] given in Theorem 3.1 can be effectively applied to prove the modularity conjecture for tadpole Nahm sums, namely, to determine for each r ≥ 2 a rational a such that q^a X_r(1,1,...,1; q) is modular.

Background

The authors derive an alternative expression for the principal character X_r(1,1,...,1; q) as a constant-term computation leading to a multiple sum involving only doubled q-Pochhammer symbols (Theorem 3.1).

They note that it is unclear whether this representation can aid in proving the modularity conjecture across ranks, raising a methodological question about the utility of this identity for resolving Conjecture 1.1.