Branching coefficient formula for the type $B_n$ Toda eigenfunction

Let $f^{B_n\operatorname{Toda}}(x|s|q)$ and $f^{A_{n-1}\operatorname{Toda}}(x|s|q)$ denote the asymptotically free eigenfunctions of the type $B_n$ and type $A_{n-1}$ $q$-Toda systems, respectively. Let $\theta=(\theta_1,\ldots,\theta_n)$ with $\theta_i\geq 0$, and write $(a)_m$ for the $q$-Pochhammer symbol. Assume the branching expansion

$$f^{B_n\operatorname{Toda}}(x_1,\ldots,x_n|s_1,\ldots,s_n|q)=\sum_{\theta_1,\ldots,\theta_n\geq 0}e^{B_n/A_{n-1}}_{\theta}(s|q)\prod_{i=1}^n x_i^{-\theta_i}f^{A_{n-1}\operatorname{Toda}}(x_1,\ldots,x_n|q^{-\theta_1}s_1,\ldots,q^{-\theta_n}s_n|q).$$

The branching coefficient formula. The coefficients are

$$e^{B_n/A_{n-1}}_{\theta}(s|q)=\prod_{k=1}^n\frac{q^{(n-k+1)\theta_k}}{(q)_{\theta_k}(q/s_k^2)_{\theta_k}}\prod_{1\leq i<j\leq n}\frac{1}{(q s_j/s_i)_{\theta_i}(q^{\theta_j-\theta_i}q s_i/s_j)_{\theta_i}}\frac{(q/s_is_j)_{\theta_i+\theta_j}}{(q/s_is_j)_{\theta_i}(q/s_is_j)_{\theta_j}}.$$

This gives an explicit branching rule expressing the type $B_n$ asymptotically free eigenfunction in terms of type $A_{n-1}$ eigenfunctions; the supplied text does not indicate whether the asserted formula has been proved or remains open.