Right-splitting morphism conjecture for Kirillov–Reshetikhin crystals

Let $\mathfrak{g}$ be a nonexceptional affine Lie algebra, let $r\in\bar{I}$, and let $a,b$ satisfy

$$a-2\ge b\ge0.$$

Suppose $\mathcal{C}'$ is a set of Kirillov–Reshetikhin crystals whose modules have been constructed and which contains $B^{r,s}$ for all $s\in\mathbb{Z}_{>0}$. Then there should be an injective $U_q(\overline{\mathfrak{g}})$-crystal morphism

$$\mathrm{rs}_{r;a,b}:B^{r,a}\otimes B^{r,b}\longrightarrow B^{r,a-1}\otimes B^{r,b+1}$$

such that, for every crystal $B$ that is a tensor product of crystals in $\mathcal{C}'$, the map

$$1_B\otimes\mathrm{rs}_{r;a,b}:B\otimes B^{r,a}\otimes B^{r,b}\longrightarrow B\otimes B^{r,a-1}\otimes B^{r,b+1}$$

is an injective $U_q(\overline{\mathfrak{g}})$-crystal morphism preserving intrinsic coenergy.

Right-splitting morphism conjecture. The stated right-splitting morphism and its extension after tensoring with any such $B$ exist with the asserted injectivity and preservation of intrinsic coenergy.

These maps are intended to split off the rightmost column in the combinatorial realization of Kirillov–Reshetikhin crystals. The supplied text gives no status evidence beyond presenting the assertion as a conjectural statement.