The decomposition conjecture for self-associate strongly typical modules

Let $\mathfrak{g}$ be the Lie superalgebra under consideration, let $U=U(\mathfrak{g})$, and let $\Pi$ denote parity shift. With $\mathfrak{p}=\mathfrak{h}\oplus\mathfrak{n}_+$, let $\lambda\in\mathfrak{h}_0^*$ be the weight of the simple $\mathfrak{h}_0$-module $V$, let $\Delta(\lambda)$ be the corresponding induced module, and set $I=\operatorname{Ann}_{U}\Delta(V)$. Let $\varphi$ be the automorphism of $U(\mathfrak{g})$ acting as $1$ on $\mathfrak{g}_0$ and as $-1$ on $\mathfrak{g}_1$, and let $(U/I)^\varphi$ denote the bimodule obtained by twisting the right action by $\varphi$.

Decomposition conjecture. Assume $\lambda$ is strongly typical and regular, and $V\cong\Pi V$. Then

$$\mathcal{L}(\Delta(\lambda),\Delta(\lambda))\cong U/I\oplus (U/I)^\varphi.$$

This describes the locally finite endomorphism bimodule in the case where the inducing module is isomorphic to its parity shift. It is proposed as a conjectural refinement of the preceding bimodule statement.