Stable one-dimensional sum conjecture for classical affine crystals

Let $G$ be one of the classical groups and let $\mu^{(1)},...,\mu^{(r)}$ and $\lambda$ index the corresponding $q$-analogue $\frak{D}_{\mu^{(1)},...,\mu^{(r)}}^{\lambda,G}(q)$ of a branching coefficient. Let $\diamondsuit_G$ denote the affine-crystal type associated with $G$, and let $X_{\mu^{(1)},...,\mu^{(r)}}^{\lambda,\diamondsuit_G}(q)$ be the corresponding stable one-dimensional sum. Stable one-dimensional sum conjecture. There is a power $q^*$, determined by the normalization of the energy or co-energy function, such that

$$\frak{D}_{\mu^{(1)},...,\mu^{(r)}}^{\lambda,G}(q)=\widetilde{K}_{\widehat{\lambda},\widehat{\mu}}^{\widehat{G},I}(q)=q^*X_{\mu^{(1)},...,\mu^{(r)}}^{\lambda,\diamondsuit_G}(q).$$

This reformulates the cited conjectural decomposition of one-dimensional sums of the non-linear classical types into sums of type $\emptyset$ one-dimensional sums; the normalization accounts for the unavoidable overall power of $q$.