The KR character formulas for types $B_n^{(1)}$ and $A_{2n-1}^{(2)}$

Let $\mathcal{Q}^{\nu}(y)$ and $\mathcal{K}^{\nu}_{D,G}(y)$ be the normalized KR-character product and associated power series. Type-$B_n$ and type-$D_n$ KR conjectures. For $B_n^{(1)}$, with $y$ specified by the type-$D_n$ identification, the formula

$$\mathcal{Q}^{\nu}(y)=\frac{\mathcal{K}^{\nu}_{D,G}(y)\prod_{a=1}^n\left(1-\prod_{k=a}^ny_k\right)^{-1}}{\displaystyle\prod_{\alpha\in\Delta_+^{D_n}}(1-e^{-\alpha})}$$

should hold for the $D_n$-characters of the KR modules. For $A_{2n-1}^{(2)}$, with the type-$D_n$ identification in the second case, the formula

$$\mathcal{Q}^{\nu}(y)=\frac{\mathcal{K}^{\nu}_{D,G}(y)\prod_{a=1}^n\left(1-y_n^{-1}\prod_{k=a}^ny_k^2\right)^{-1}}{\displaystyle\prod_{\alpha\in\Delta_+^{D_n}}(1-e^{-\alpha})}$$

should hold for the $D_n$-characters of the KR modules. These are equivalent reformulations of the canonical-solution conjecture for the two stated affine types; no resolution is supplied in the source.