The injectivity conjecture for strongly typical regular central quotients

Let $\mathfrak{g}$ be the Lie superalgebra under consideration, let $U=U(\mathfrak{g})$, and, with $\mathfrak{p}=\mathfrak{h}\oplus\mathfrak{n}_+$, let $\lambda\in\mathfrak{h}_0^*$ be the weight of the simple $\mathfrak{h}_0$-module $V$. Write $\Delta(V)$ for the corresponding induced module and $I=\operatorname{Ann}_{U}\Delta(V)$. The adjoint module $(U/I)^{\mathrm{ad}}$ is $U/I$ with the adjoint $\mathfrak{g}$-action.

Injectivity conjecture. Assume $\lambda$ is strongly typical and regular. Then $(U/I)^{\mathrm{ad}}$ is a direct sum of injective finite-dimensional modules.

The conjecture concerns the structure of the adjoint module of a central quotient of the enveloping algebra. The paper notes that the corresponding assertion for $\mathcal{L}(\Delta(\lambda),\Delta(\lambda))^{\mathrm{ad}}$ follows from exactness, and that the conjecture holds when $V\not\cong\Pi V$; the general case remains conjectural.