Dirac cohomology conjecture for irreducible restricted rational Cherednik modules

Let $\overline{{\bf H}}$ be the restricted rational Cherednik algebra, let $\mathfrak{h}$ be the reflection representation of $W$, let $\widetilde{W}$ be the associated pin double cover, and let $\chi$ be the spin module character. For an irreducible $\overline{{\bf H}}$-module $\overline{L}(\sigma)$, write $H^{\bullet}(\mathfrak{h}^*,\overline{L}(\sigma))$ and $H_{\bullet}(\mathfrak{h},\overline{L}(\sigma))$ for its cohomology and homology, and let $H_D(\overline{L}(\sigma))$ denote its Dirac cohomology. Dirac cohomology conjecture. For irreducible $\overline{{\bf H}}$-module $\overline{L}(\sigma)$, there are $\widetilde{W}$-module isomorphisms

$$H_D(\overline{L}(\sigma)) \cong H^{\bullet}(\mathfrak{h}^*,\overline{L}(\sigma)) \otimes \chi \cong H_{\bullet}(\mathfrak{h},\overline{L}(\sigma)) \otimes \chi.$$

The claim is stated by analogy with the case $t=1$; the supplied text does not establish it, so its general status remains open.