Ext-vanishing conjecture for thick and local Demazure modules

Let $\Lambda$ be a level-one fundamental weight, let $b,c\in P$, let $m\in\mathbf Q$, and let $p>0$. Let $D_b$ and $D_c$ be thick Demazure modules, let $\mathring{\mathbb D}_b$ and $\mathring{\mathbb D}_c$ be local Demazure slices, and let $\mathbf C_{\Lambda_0+m\delta}$ denote the corresponding one-dimensional character module. Ext-vanishing conjecture. One has

$$\operatorname{Ext}^p_{(\widetilde{\mathfrak b}^-,\widetilde{\mathfrak h})}\left(D_b\otimes L(\Lambda),\mathring{\mathbb D}_c^\vee\otimes\mathbf C_{\Lambda_0+m\delta}\right)=\{0\},$$

and

$$\operatorname{Ext}^p_{(\widetilde{\mathfrak b}^-,\widetilde{\mathfrak h})}\left(\mathring{\mathbb D}_b\otimes L(\Lambda),D_c^\vee\otimes\mathbf C_{\Lambda_0+m\delta}\right)=\{0\}.$$

These vanishings are proposed as part of the homological explanation for the remaining decompositions of the paper's main theorem; the supplied text gives no resolution.