Modular Nahm Sums for the Inverse Cartan Matrix of Type $D_r$

For $r\ge 3$ we denote by $\mathcal{C}(D_r)$ the Cartan matrix of type $D_r$. Recently, Sun and Wang conjectured a Rogers–Ramanujan type identity for the Nahm sum associated with $\mathcal{C}(D_r{)}^{-1}$ and the zero vector. They further conjecture that there exist $r-1$ companion modular Nahm sums associated with nonzero vectors. We partially prove this conjecture by constructing $\lfloor (r+4)\mathrm{/}2\rfloor$ modular Nahm sums for $\mathcal{C}(D_r{)}^{-1}$. To prove their modularity, we utilize the method of Bailey pairs to establish various Rogers–Ramanujan type identities. In particular, we confirm their conjectural identity.