Dirac cohomology conjecture for irreducible rational Cherednik algebra modules

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

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

The preceding argument proves the corresponding assertion under the hypotheses of the theorem in the paper; the statement is presented as the section's concluding claim, and its general resolution is not supplied here.