Calinescu–Milas–Penn conjecture on modularity of principal tadpole Nahm sums

Let $T_r=A_{2r}/\mathbb{Z}_2$ be the tadpole Dynkin diagram, and let $\mathbf{T}_r$ be its Cartan matrix. For $\mathbf{x}_r=(x_1,\ldots,x_r)$, define the tadpole Nahm sum

$$\mathcal{T}_r(\mathbf{x}_r;q)=\sum_{\mathbf{n}_r=(n_1,\ldots,n_r)\in\mathbb{Z}^r}\frac{q^{\frac{1}{2}\mathbf{n}_r\mathbf{T}_r\mathbf{n}_r^\mathsf{T}}x_1^{n_1}\cdots x_r^{n_r}}{(q)_{n_1}\cdots(q)_{n_r}}.$$

The sum is principal when $x_1=\cdots=x_r=1$. Calinescu–Milas–Penn's conjecture. The principal tadpole Nahm sum $\mathcal{T}_r(1,\ldots,1;q)$, shifted by a rational power of $q$, is modular. This conjecture generalizes the confirmed modularity results in ranks two through five; its general status is not resolved in the supplied context.