Schilling–Shimozono fermionic formula conjecture for affine crystals

Let $B$ be a tensor product of Kirillov–Reshetikhin crystals, let $\diamondsuit\in\{(1),(2),(1,1)\}$ denote the kind of the nonexceptional affine family, and let $\overline{X}_{\lambda,B}^{\diamondsuit}(q)$ be the corresponding one-dimensional sum. Write $|B|=\sum_i r_i s_i$, let $|\lambda|$ be the size of the partition $\lambda$, let $\mathcal{P}_n$ be the partitions with at most $n$ parts, let $\mathcal{P}_n^{\diamondsuit}$ be the partitions of kind $\diamondsuit$, and let $c_{\delta\lambda}^{\nu}$ be the Littlewood–Richardson coefficient.

Schilling–Shimozono conjecture. For $\diamondsuit\in\{(1),(2),(1,1)\}$,

$$\overline{X}_{\lambda,B}^{\diamondsuit}(q)=q^{\frac{|B|-|\lambda|}{|\diamondsuit|}}\sum_{\nu\in\mathcal{P}_n}\sum_{\delta\in\mathcal{P}_n^{\diamondsuit}}c_{\delta\lambda}^{\nu}\,\overline{X}_{\nu,B}^{\varnothing}\left(q^{\frac{2}{|\diamondsuit|}}\right).$$

The conjecture gives a uniform relation between one-dimensional sums for the nonexceptional affine families and the type $A$ family. The paper states that its main purpose is to establish this conjecture; the supplied text does not explicitly provide a resolution status.