Universal cyclic thick Demazure module conjecture

Let $P$ be the weight lattice, let $D_c$ and $\mathbb D_c$ be the thick and local Demazure modules, and define

$$R_c=\bigotimes_{i=1}^n\mathbf C[X_{i,1},\ldots,X_{i,m_i}]^{\mathfrak S_{m_i}},\qquad m_i=(\alpha_i^\vee,c)^\text{\tiny$\!{}^\maltese$},$$

with $\deg X_{i,j}=1$. Let $\widetilde{\mathfrak b}^-$ be the relevant affine negative Borel algebra, $h_c^0$ the quantity from the source's equation (limeval), and let $\mathbf C_0$ denote the specialization of $R_c$ at the origin. Universal cyclic module conjecture. For every $c\in P$, there exists a maximal universal cyclic $\widetilde{\mathfrak b}^-$-module $\mathsf D_c$ such that: (1) $\mathsf D_c$ maps surjectively onto $D_c$ as a $\widetilde{\mathfrak b}^-$-module; (2) $\operatorname{End}_{\widetilde{\mathfrak b}^-}(\mathsf D_c)\cong R_{-c}$ and $\mathsf D_c$ is free as an $R_{-c}$-module; and (3) $h_c^0\,\mathsf{gch}\,\mathsf D_c=\mathsf{gch}\,D_c$. Moreover, if $\mathring{\mathbb D}_c=\mathbf C_0\otimes_{R_c}\mathbb D_c$, then

$${\text{\rm dim}}_{\mathbb C}\,_{\mathbf C}\operatorname{Ext}^p_{(\widetilde{\mathfrak b}^-,\widetilde{\mathfrak h})}\left(\mathsf D_b\otimes_{\mathbf C}\mathbf C_{m\delta},\mathring{\mathbb D}_c^\vee\right)=\delta_{p,0}\delta_{b,c}\delta_{m,0}.$$

These conjectures propose universal modules whose endomorphism rings control thick Demazure modules and whose local slices satisfy an orthogonality relation. The source notes that the analogous orthogonality conjecture for types $\mathsf{ADE}$ was essentially proved, while the general assertion remains conjectural.