The tensor-product realization conjecture for irregular minimal affinizations

Let $\lie g$ be of type $D$ or $E$, with the decomposition used in the paper into subdiagrams $I_1$, $J$, and $I_2$, where $J$ is of type $D_4$. Let $\lambda\in P^+$ and let $\gb\lambda\in\cal P_q^+$ be such that $V_q(\gb\lambda)$ is a minimal affinization of $V_q(\lambda)$. Let $N$ be the $\lie g[t]$-submodule of

$$L(\lambda^{I_1})\otimes L(\gb\lambda^J)\otimes L(\lambda^{I_2})$$

generated by the top weight space. Tensor-product realization conjecture. There is an isomorphism

$$L(\gb\lambda)\cong N.$$

A proposition establishes only that $L(\gb\lambda)$ projects onto this submodule; the conjecture strengthens the projection to an isomorphism in the irregular case, where the number of minimal-affinization classes is not uniformly bounded.