Hoshino–Shiraishi formulas for the branching coefficients in ranks two and three

Let $s=(s_1,\ldots,s_n)$, let $\theta=\{\theta_{ij}\}_{1\leqslant i<j\leqslant n}$ be nonnegative integers, and define

$$\phi_i=\sum_{j=1}^{i-1}\theta_{ji}+\sum_{j=i+1}^n\theta_{ij}.$$

The branching coefficients $e_n(s;\theta;q,t)$ are defined by the branching expansion described in the source. Hoshino–Shiraishi conjectures. For rank two,

$$e_2(s_1,s_2;\theta_{12};q,t)=\frac{(t,t/s_1,t/s_2,q^{\theta_{12}+1}/(ts_1s_2);q)_{\theta_{12}}}{(q,q/s_1,q/s_2,q^{\theta_{12}}/(s_1s_2);q)_{\theta_{12}}}\left(\frac qt\right)^{\theta_{12}},$$

and for rank three, the coefficient is the displayed product formula in the source, with $\theta_{ji}=\theta_{ij}$ for $1\leqslant i<j\leqslant3$. These formulas conjecturally determine the first nontrivial branching coefficients for the folded asymptotically free solutions.