Odd-length denominator formula for type C affine quantum algebras

Let $m,p\in 2\mathbb{Z}_{\geq 1}+1$, let $k<n$, and let $\mathfrak{g}=C_n^{(1)}$. Write $d_{k^m,k^p}(z)$ for the denominator polynomial of the corresponding normalized $R$-matrix, and let $q_{\mathfrak{s}}$ denote the parameter associated with the short root. Odd-length denominator formula. We have

$$d_{k^m,k^p}(z)=\prod_{t=0}^{\min(m,p)-1}\prod_{s=1}^{k}\bigl(z-(-q_{\mathfrak{s}})^{|m-p|+2(s+t)}\bigr)\bigl(z-(-q_{\mathfrak{s}})^{2n+2-2k+|m-p|+2(s+t)}\bigr).$$

This gives the denominator explicitly for the stated odd values of $m$ and $p$ in type $C_n^{(1)}$; the surrounding denominator theorem supplies related formulas in other cases, while this particular case is presented as a conjecture.