Zagier-duality modularity conjecture for type $D_k$ Cartan matrices

Let $\mathcal{C}(D_k)$ be the Cartan matrix of the Dynkin diagram of type $D_k$, and let $f_{A,b,C}(q)$ denote the corresponding Nahm sum. Zagier-duality conjecture. For every $k\geq 3$, the Nahm sum

$$f_{\frac{1}{2}\mathcal{C}(D_k),0,(1-k)/24}(q)$$

is modular. This proposed identity is the dual counterpart of the modularity theorem proved in the paper for $f_{2\mathcal{C}(D_k)^{-1},0,-1/24}(q)$; its validity for all $k\geq 3$ remains open.