Extension conjecture for partial Verma functors

Let $\chi$ be semisimple and lie in a Cartan subalgebra $\mathfrak{h}$, and let $T$ be the corresponding torus. The $T$-action gives graded module categories $\mathcal{U}^{[r]}_0(\mathfrak{g})\text{-mod}^{gr}$ and $\mathcal{U}^{[r+1]}_\chi(\mathfrak{g})\text{-mod}^{gr}$. For a simple module $L$ with weight $\lambda$, the assignment is $(L,\lambda)\mapsto Z^{(r)}_\chi(\lambda)\otimes L$. Extension conjecture. The assignment on simples, and similarly on modules obtained by restriction from $\mathcal{U}^{[r]}(\mathfrak{g})$, extends to a functor

$$F:\mathcal{U}^{[r]}_0(\mathfrak{g})\text{-mod}^{gr}\longrightarrow\mathcal{U}^{[r+1]}_\chi(\mathfrak{g})\text{-mod}^{gr}.$$

This proposes a graded extension of the partial Verma-module construction to the relevant module categories. The source does not provide evidence resolving the conjecture.