Dual-morphism conjecture for the explicit higher-degree construction

Let $L$ be the graded Lie superalgebra under consideration, let $V$ and $W$ be $L_0$-modules, and let $M(V)$ and $M(W)$ be their generalized Verma modules. Suppose that a degree-$d$ morphism $\varphi:M(V)\rightarrow M(W)$ is associated to

$$\Phi:=\sum_{T,I}{\partial_T\omega_I}\otimes \theta^T_I,$$

where $\theta^T_I\in\operatorname{Hom}(V,W)$. For $T\in[5]^k$, define $\ell(T)=k$, and let $(\theta^T_I)^*$ denote the pull-back map $W^*\rightarrow V^*$.

Dual-morphism conjecture. The linear map $\psi:M(W^*)\rightarrow M(V^*)$ associated to

$$\Psi:=\sum_{T,I}\partial_T\omega_I\otimes (-1)^{\ell(T)}(\theta^T_I)^*$$

is also a morphism of Verma modules.

This is the explicit all-degree formulation of the preceding duality conjecture, with the construction established in the paper for degrees at most three. The supplied context does not state whether the all-degree assertion has been resolved.