Higher baby Verma quotient conjecture

Let $G$ be a reductive algebraic group, let $\chi\in\mathfrak g^*$, and let $U_\chi^{[r]}(G)$ be the corresponding higher universal enveloping algebra. For an irreducible $\operatorname{Dist}(G_{(r)})$-module $N$ with weight extending to $\lambda_r\in\Lambda_\chi^r$, let $\widehat{U_\chi^{[r]}(B)}$ be the higher Borel subalgebra and define

$$Z_\chi^{[r]}(N,\lambda_r)\coloneqq U_\chi^{[r]}(G)\otimes_{\widehat{U_\chi^{[r]}(B)}}N.$$

Higher baby Verma quotient conjecture. Every irreducible $U_\chi^{[r]}(G)$-module is a homomorphic image of $Z_\chi^{[r]}(N,\lambda_r)$ for some irreducible $\operatorname{Dist}(G_{(r)})$-module $N$ and $\lambda_r\in\Lambda_\chi^r$ extending the weight of $N$, where $N$ has the $\widehat{U_\chi^{[r]}(B)}$-module structure supplied by the preceding conjecture.

This is the higher analogue of the standard assertion that every irreducible Lie algebra module is a quotient of a baby Verma module. The source says that the conjecture is proved in the sequel.