Compact Chern character and weighted orbital integral conjecture

Let $G$ be a $p$-adic group, let ${\rm Sm}$ be the category of smooth modules and ${\rm Adm}\subset {\rm Sm}$ the subcategory of admissible modules. Let $HH_0^c({\rm Sm})$ and $HH_0(\overline{\rm Sm})$ denote compactly supported and compactified Hochschild homology, let $\mathcal K_G^c$ be the subspace of weightless measures supported on compact elements, and let $\bar{ch}$ be the Chern character map. Compact Chern character and weighted orbital integral conjecture. There is a canonical isomorphism

$$\mathcal K_G^c\cong {\rm Im}\bigl(HH_0^c({\rm Sm})\to HH_0(\overline{\rm Sm})\bigr),$$

and, for $\rho\in{\rm Adm}$ and compact regular semisimple $g\in G$,

$$WO_g(\bar{ch}(\rho))=\chi_\rho(g).$$

This connects compactly supported Hochschild homology and Chern characters with character values; the supplied text does not state whether these assertions are known beyond the cases discussed in the paper.