General modularity conjecture for tadpole Nahm sums

Let $\chi_0({\bf 1})$ be the Nahm sum associated to the tadpole Cartan matrix $T_n$, with $n\geq 2$. For even $n=2k\geq 2$, define

$$R_{2k,i}(\tau)=\sum_{n\in\mathbb{Z}}(-1)^nq^{(k+1)(n-\frac{2i-1}{4(k+1)})^2}.$$

For odd $n=2k-1\geq 3$, use the derivative theta functions $(\partial\Theta)_{i,\frac{2k+1}{2}}$. Let $\frak{f}(\tau)$ and $\widetilde{\mathcal{W}}_D$ denote the functions and modified Wronskian appearing in the source.

General tadpole modularity conjecture. For even $n=2k\geq 2$,

$$q^{a_k}\chi_0({\bf 1})=\frac{\frak{f}(\tau)^{2k}\widetilde{\mathcal{W}}_D(R_{2k,1},\ldots,R_{2k,k})}{\eta(\tau)^{k(2k-1)}},$$

where $a_k=-\frac{k(1+4k)}{48(1+k)}$. For odd $n=2k-1\geq 3$,

$$q^{a_k}\chi_0({\bf 1})=\frac{\frak{f}(\tau)^{2k-1}\widetilde{\mathcal{W}}_D((\partial\Theta)_{1,\frac{2k+1}{2}},\ldots,(\partial\Theta)_{k-1,\frac{2k+1}{2}})}{\eta(\tau)^{(k-1)(2k-1)}},$$

where $a_k=\frac{-1+6k-8k^2}{96k+48}$. More precisely, $q^{a_k}\chi_0({\bf 1})$ is a modular function that is a component of a $3k$-dimensional vector-valued modular function under $\Gamma(1)$.

The claim is based on numerical evidence and proposes an explicit modular description for all tadpole ranks $n\geq 2$. The source does not report a proof or a refutation.