Folklore modularity conjecture for ADET Cartan-matrix Nahm sums

Let $X$ and $Y$ be Dynkin diagrams of ADET type, with Cartan matrices $\mathcal{C}(X)$ and $\mathcal{C}(Y)$, ranks $r(X)$ and $r(Y)$, and Coxeter numbers $h(X)$ and $h(Y)$. Define

$$M(X,Y)=\mathcal{C}(X)\otimes\mathcal{C}(Y)^{-1},\qquad C(X,Y)=-\frac{1}{24}\frac{r(X)r(Y)h(X)}{h(X)+h(Y)}.$$

Here $\otimes$ denotes the Kronecker product. Folklore modularity conjecture. The Nahm sum associated with $(M(X,Y),0,C(X,Y))$ is modular. This conjecture provides a systematic source of modular Nahm sums from Cartan matrices and is motivated by fermionic character formulas for coset two-dimensional conformal field theories. Several special cases are known, but the general ADET statement is not resolved in the supplied text.