Vanishing Ext conjecture for anti-spherical discrete series

Let $\mathbb H$ be the graded Hecke algebra, let $\Delta$ be its set of simple roots, and for $I\subset\Delta$ let $\widetilde{\mathbb H}_I$ be the corresponding graded Hecke algebra. Let $X'$ be an anti-spherical discrete series of $\mathbb H$, and let $Y'$ be an anti-spherical discrete series of $\widetilde{\mathbb H}_I$. Vanishing Ext conjecture. For every $i\geq 1$,

$$\operatorname{Ext}^i_{\widetilde{\mathbb H}_I}(X'|_{\widetilde{\mathbb H}_I},Y')=0.$$

This conjecture proposes that the Ext-branching vanishing remains valid after dropping the regularity condition, provided both discrete series are anti-spherical. The preceding discussion shows that vanishing can fail when the anti-spherical condition is dropped; the status of the stated generalization is not resolved in the source.