Quantum trace compatibility for spin generators

Let $\iota:{\rm End}_{U_q(\mathfrak{so}_{2n+1})}(S^{\otimes(m-1)})\rightarrow{\rm End}_{U_q(\mathfrak{so}_{2n+1})}(S^{\otimes m})$ be given by $\iota(\mathsf W)=\mathsf W\otimes{\rm id}_S$. Quantum trace conjecture. For $0\leq k\leq n$ and $\mathsf W\in{\rm End}_{U_q(\mathfrak{so}_{2n+1})}(S^{\otimes(m-1)})$,

$$\operatorname{Tr}_q(\mathsf X_{m-1}^{(k)}\circ\iota(\mathsf W))=(-1)^{n(k+1)+\binom{n-k}{2}}\prod_{t=1}^{n-k}\frac{[n+1-t][n+t]}{[t]^2}\operatorname{Tr}_q(\mathsf W).$$

This trace identity is presented as an elementary conjectural ingredient in the characterization of the spin link polynomial and is known immediately for $k=0$ and $k=n$; the general case remains open in the source.