Generalized Dynkin-diagram modularity conjecture for Nahm quadruples

Let $X$ and $Y$ be Dynkin diagrams of type $ABCDEFGT$, with Cartan matrices $C(X)$ and $C(Y)$. Let $D(X)$ be the diagonal matrix whose $j$-th diagonal entry is $\alpha_j^2/\beta^2$, where $\alpha_j$ is the $j$-th simple root and $\beta$ is a short root. Define

$$A(X,Y)=C(X)\otimes C(Y)^{-1}, \qquad D(X,Y)=D(X)\otimes D(Y),$$

where $\otimes$ is the Kronecker product. Set $C(X,Y)=-c(X,Y)/24$, with

$$c(X,Y)=\frac{\operatorname{tr}(D(X,Y))h(X)}{h(X)+h(Y)},$$

where $h(-)$ is the Coxeter number. Generalized modularity conjecture. The quadruple $(A(X,Y),0,C(X,Y),D(X,Y))$ is a modular quadruple. The statement has been disproved: Wang found counterexamples to Zagier's duality conjecture and its generalization, so the supplied status evidence marks this conjecture as refuted.