The conjectural Kirillov–Reshetikhin module family

Let $X_n^{(1)}$ be a non-twisted quantum affine algebra, with $1\le r\le n$ and $s\ge1$. Let $t_r$ be the root-data parameter appearing in the source, let $\overline{P}^+$ be the dominant finite-type weight lattice, and let $V(\lambda)$ denote the irreducible $U_q(X_n)$-module of highest weight $\lambda$. For each pair $(r,s)$, consider a finite-dimensional $U'_q(X_n^{(1)})$-module $W^{(r)}_s$ and its crystal base $B^{r,s}$; write $\operatorname{ch}W^{(r)}_s$ for its character, and let $M(W^{(r)}_s,\lambda,q)$ be the fermionic multiplicity form defined in the source.

Conjectural Kirillov–Reshetikhin module family. For every $(r,s)$, there exists an irreducible finite-dimensional $U'_q(X_n^{(1)})$-module $W^{(r)}_s$ such that:

1. $B^{r,s}$ is a finite crystal of level $\left\lceil\frac{s}{t_r}\right\rceil$, perfect when $s/t_r$ is an integer and non-perfect otherwise.
2. As a $U_q(X_n)$-module,

$$W^{(r)}_s=\bigoplus_{\lambda\in\overline{P}^+}M(W^{(r)}_s,\lambda,q=1)V(\lambda).$$

3. If $Q^{(r)}_s=\operatorname{ch}W^{(r)}_s$, then $Q^{(r)}_s$ satisfies the $Q$-system defined in the source. The family is intended to provide finite-dimensional modules and crystals uniformly for all non-twisted quantum affine types; the source presents it as conjectural and does not state a resolution.