The Moura–Pereira conjecture on modules associated with minimal affinizations

Let $\lie g$ be a simple Lie algebra, let $P^+$ denote its dominant integral weights, and let $M(\lambda)$ and $T(\lambda)$ be the graded $\lie g[t]$-modules defined in the paper for $\lambda\in P^+$. For $\gb\lambda\in\cal P_\mathbb A^\times$, let $V_q(\gb\lambda)$ be a minimal affinization of the irreducible $U_q(\lie g)$-module $V_q(\lambda)$, and let $L(\gb\lambda)$ denote its classical limit. Moura–Pereira's conjecture. For every $\lambda\in P^+$,

$$M(\lambda)\cong T(\lambda).$$

Moreover, if $\overline{\rm supp}(\lambda)$ is of type $A$ and $V_q(\gb\lambda)$ is a minimal affinization of $V_q(\lambda)$, then

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

The conjecture identifies the modules defined by generators and relations with the graded modules arising from classical limits of minimal affinizations. The source gives no resolution status; the claim is presented as the main conjecture of the cited work.