The $T_k(\lambda)$–$M_k(\lambda)$ conjecture for regular multiple minimal affinizations

Let $\lie g$ be of type $D$ or $E$, let $i_0$ be its unique triply connected node, and let $\lambda\in P^+$ be supported on the three connected components of $I\setminus\{i_0\}$. For $k\in\{1,2,3\}$, let $M_k(\lambda)$ and $T_k(\lambda)$ be the modules defined in the paper. $T_k(\lambda)$–$M_k(\lambda)$ conjecture. For every $k\in\{1,2,3\}$,

$$T_k(\lambda)\cong M_k(\lambda).$$

The modules admit surjections $M_k(\lambda)\twoheadrightarrow L(\gb\lambda)\twoheadrightarrow T_k(\lambda)$ in the relevant minimal-affinization setting, and the conjecture asserts that the two endpoint modules coincide. The source gives no resolution.