Large-rank restriction conjecture for minimal affinizations

Let $\diamondsuit$ be one of the four allowed diagram shapes and let $\{\mathfrak{g}_n\}$ be a nonexceptional family of affine algebras associated with $\diamondsuit$. Let $\mathcal{P}^{\diamondsuit}$ be the set of partitions whose Ferrers diagrams can be tiled by $\diamondsuit$, let $W^\lambda$ be the minimal affinization associated with a partition $\lambda$, and let $c^\tau_{\lambda\mu}$ denote the Littlewood–Richardson coefficient, equivalently the stable type $A$ tensor multiplicity. Large-rank restriction conjecture. For $n$ sufficiently large and partitions representing dominant weights as above,

$$W^\lambda\cong\bigoplus_\tau\left(\sum_{\mu\in\mathcal{P}^{\diamondsuit}}c^\tau_{\lambda\mu}\right)V^\lambda$$

as $U_q(\overline{\mathfrak{g}})$-modules. This conjecture describes the stable restriction from type $A$ through the partition shapes associated with the affine family; the source gives no resolution status.