The $(\mathrm{C}_n,\mathrm{B}_n)$ branching rule

Let $m,r$ be nonnegative integers. For the partition statistics and coefficient functions appearing below, and for the universal Koornwinder-type polynomial $\tilde{K}$, one has

$$P_{(m^r)}(q,t)=\sum_{\substack{\lambda\subset (m^r) \\ \operatorname{2\text{-}core}(\lambda)=0}} f_{\lambda}\big(q^{-m},q^r/T;q,t\big)\tilde{K}_{(m^r)-\lambda}\big(q,t,T;\pm q^{1/2},\pm t^{1/2}\big).$$

For nonnegative integers $m,r,n$ with $r\leqslant n$, and $x=(x_1,\ldots,x_n)$, one also has

$$P_{(m^r)}(x^{\pm};q,t)=\sum_{\substack{\lambda\subset (m^r) \\ \operatorname{2\text{-}core}(\lambda)=0}} f_{\lambda}\big(q^{-m},q^{-(n-r)};q,t\big)P^{(\mathrm{C}_n,\mathrm{B}_n)}_{(m^r)-\lambda}(x;q,t,t),$$

where

$$f_{\lambda}(w,z;q,t):=\left(\frac qt\right)^{|\lambda|/2}q^{2\hat n^{\mathrm{o}}(\lambda')-2\hat n^{\mathrm{e}}(\lambda')}t^{n^{\mathrm{e}}(\lambda)-n^{\mathrm{o}}(\lambda)}\frac{C^0_{\lambda}(w;q,t)}{C^0_{\lambda}(qw/t;q,t)}\frac{C^{-,\mathrm{e}}_{\lambda}(t;q,t)}{C^{-,\mathrm{o}}_{\lambda}(q;q,t)}\frac{C^{+,\mathrm{e}}_{\lambda}(qw^2z^2/t^2;q,t)}{C^{+,\mathrm{o}}_{\lambda}(w^2z^2/t;q,t)}.$$

The $(\mathrm{C}_n,\mathrm{B}_n)$ branching-rule conjecture. The two displayed identities hold.