The arbitrary-level monoidal categorification conjecture

Let $\mathfrak g$ be a simple Lie algebra, let $I$ be its index set, let $\ell$ be a positive integer, and let ${\cal C}_\ell$ be the corresponding monoidal subcategory with Grothendieck ring $R_\ell$. Let ${\cal A}_\ell={\cal A}(\widetilde{B}_\ell)$ be the cluster algebra with initial variables $x_{(i,k)}$, and let $r(i,k)$ be the integers defined in the paper. Write $W^{(i)}_{k,r}$ for the Kirillov–Reshetikhin module with node $i$, level $k$, and spectral parameter $q^r$.

Arbitrary-level monoidal categorification conjecture. The assignment

$$x_{(i,k)}\longmapsto [W^{(i)}_{k,r(i,k)}]$$

extends to a ring isomorphism $\iota:{\cal A}_\ell\to R_\ell$. Under this identification, ${\cal C}_\ell$ is a monoidal categorification of ${\cal A}_\ell$.

This generalizes the level-one conjecture to arbitrary $\ell$. The cluster algebra is often of infinite type, and the source presents the assertion as a main conjecture rather than a theorem.