Predicted socles and cosocles of exotic standard and costandard objects

Let $G$ be the reductive group and let $\boldsymbol{X}$ be its weight lattice. For $\lambda \in \boldsymbol{X}$, write $\operatorname{dom}(\lambda)$ for the dominant representative, let $\delta_\lambda$ be the grading shift attached to $\lambda$, let $\mathsf{m}(\lambda)$ be the relevant minimal-weight operation, and let $\mathfrak{E}_\mu$ denote the corresponding exotic simple object. Let $\hat\Delta_\lambda$ and $\hat\nabla_\lambda$ be the exotic standard and costandard objects. Predicted exotic socle and cosocle statement. The socle of $\hat\Delta_\lambda$ is isomorphic to

$$\mathfrak{E}_{\accentset{-}{\mathsf{m}}(\lambda)}\langle -(2\rho^\vee,\operatorname{dom}(\lambda))+\delta_\lambda\rangle,$$

and the cokernel of its inclusion into $\hat\Delta_\lambda$ contains no composition factor of the form $\mathfrak{E}_\mu\langle m\rangle$ with $\mu \in -\boldsymbol{X}_{\mathrm{min}}$. The cosocle of $\hat\nabla_\lambda$ is isomorphic to

$$\mathfrak{E}_{\accentset{-}{\mathsf{m}}(\lambda)}\langle (2\rho^\vee,\operatorname{dom}(\lambda))-\delta_\lambda\rangle,$$

and the kernel of the quotient map onto this cosocle contains no composition factor of the form $\mathfrak{E}_\mu\langle m\rangle$ with $\mu \in -\boldsymbol{X}_{\mathrm{min}}$. The statement predicts the socles and cosocles of exotic standard objects; the source notes that it is confirmed for $G=\mathrm{SL}_2$, while the general case is left as a prediction.