The $A_{2n}^{(2)}$ KR character formula in type $B_n$

Let $X_N^{(r)}=A_{2n}^{(2)}$, so that $\mathfrak{g}_0=B_n$, and let $\mathcal{Q}^{\nu}(y)$ and $\mathcal{K}^{\nu}_{D,G}(y)$ be the normalized KR-character product and associated power series. Type-$B_n$ KR conjecture. The formula

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

should hold for the normalized $B_n$-characters of the KR modules. This is the separate formulation required in the twisted $A_{2n}^{(2)}$ case because the usual denominator relation fails; no resolution is supplied.