The diagonal residue product conjecture for affine Weyl groups

Let $S$ be one of the sets of nodes associated to an affine simply-laced type, let $T$ be its complementary set, let $\alpha_0$ be the affine root, and write $\alpha_0|_S$ and $\alpha_0|_T$ for its restrictions. Let $R_1({\mathbf{x}}_S)$ denote the diagonal part of the residue, viewed as a power series in ${\mathbf{x}}_S^{\alpha_0|_S}$, and let $\operatorname{ht}$ denote height. For the listed affine types and sets $S$, define $\lambda$ by the table in the claim.

Diagonal residue product conjecture. We have

$$R_1({\mathbf{x}}_S)=\prod_{m=0}^{\infty} (1-q^{(m+1)(\operatorname{ht}(\alpha_0|_S)-\operatorname{ht}(\alpha_0|_T))} {\mathbf{x}}^{(2m+2) \alpha_0|_S})^{-|T|}$$ $$\qquad\cdot(1-q^{(m+1)(\operatorname{ht}(\alpha_0|_S)-\operatorname{ht}(\alpha_0|_T))+1} {\mathbf{x}}^{(2m+2) \alpha_0|_S})^{-|T|}$$ $$\qquad\cdot(1-q^{(m+1/2)(\operatorname{ht}(\alpha_0|_S)-\operatorname{ht}(\alpha_0|_T))} {\mathbf{x}}^{(2m+1) \alpha_0|_S})^{-\lambda}$$ $$\qquad\cdot(1-q^{(m+1/2)(\operatorname{ht}(\alpha_0|_S)-\operatorname{ht}(\alpha_0|_T))+1} {\mathbf{x}}^{(2m+1) \alpha_0|_S})^{-\lambda}.$$

Here $\lambda$ is $1$ for $\widetilde{A}_n$ with $n$ odd and for $\widetilde{D}_n$ with $n$ odd; for $\widetilde{D}_n$ with $n$ even it is $0$ when $S=\{1,2,4,6,\ldots,n-2,n,n+1\}$ and $3$ when $S=\{3,5,7,\ldots,n-1\}$; for $\widetilde{E}_6$ it is $0$ for $S=\{1,3,5,7\}$ and $1$ for $S=\{2,4,6\}$; for $\widetilde{E}_7$ it is $0$ for $S=\{1,3,4,5,7\}$ and $2$ for $S=\{2,6,8\}$; and for $\widetilde{E}_8$ it is $1$ for $S=\{1,5,7,9\}$ and $0$ for $S=\{2,3,4,6,8\}$.

The diagonal residue is expected to admit an infinite-product description in terms of function-field zeta factors, complementing the already computed part $R_0$. The source states this as a conjecture for the listed affine simply-laced types, but gives no resolution evidence.