Milas–Wang conjecture on a Wronskian formula for principal tadpole Nahm sums

Let $T_r=A_{2r}/\mathbb{Z}_2$ be the tadpole Dynkin diagram, and let $\mathcal{T}_r(\mathbf{x}_r;q)$ denote its tadpole Nahm sum. The sum $\mathcal{T}_r(1,\ldots,1;q)$ is the principal tadpole Nahm sum. A generalized theta series is understood in the sense used for the length-$r$ sequence appearing in the conjectured formula, and a Wronskian is the determinant formed from that sequence and its derivatives. Milas–Wang's conjecture. The principal tadpole Nahm sum $\mathcal{T}_r(1,\ldots,1;q)$, shifted by a rational power of $q$, can be explicitly expressed as the product of a modular infinite product and the Wronskian of a length-$r$ sequence of generalized theta series. The paper presents this as a prediction intended to give a complete resolution of the Calinescu–Milas–Penn modularity conjecture, while the supplied context does not state that it has been proved.