The bimodule direct-summand conjecture for strongly typical regular 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 $V$. Let $\Delta(\lambda)$ be the corresponding induced module, write $\mathcal{L}(\Delta(\lambda),\Delta(\lambda))$ for the bimodule of locally finite homomorphisms, and set $I=\operatorname{Ann}_{U}\Delta(V)$.

Bimodule direct-summand conjecture. Assume $\lambda$ is strongly typical and regular. Then the bimodule $U/I$ is a direct summand of

$$\mathcal{L}(\Delta(\lambda),\Delta(\lambda)).$$

This stronger bimodule statement obviously implies the conjecture about the adjoint module being a direct sum of injective finite-dimensional modules. It is presented as conjectural in the source.