Penn–Calinescu–Wang modularity conjecture for tadpole Nahm sums

Let $T_r$ be the tadpole Cartan matrix and let

$$\chi_0(x_1,\dots,x_r)=\sum_{n=(n_1,\dots,n_r)\in\mathbb{Z}_{\geq 0}^r}\frac{q^{\frac12 n^{\mathrm T}T_r n}x_1^{n_1}\cdots x_r^{n_r}}{(q)_{n_1}\cdots(q)_{n_r}}$$

be the associated generalized tadpole Nahm sum. The character with all variables specialized to $1$ is $\chi_0(1,\dots,1)$.

Penn–Calinescu–Wang modularity conjecture. The character $q^a\chi_0(1,\dots,1)$ is modular for some rational number $a$.

The rank-two case $r=2$ was proved in the cited work, while the paper investigates the rank-three case and provides further evidence for the conjecture.