Weightless functions and invariant distributions conjecture

Let $G$ be a $p$-adic group, let $\mathcal K\subset\mathcal H(G)$ be the space of weightless functions, let $\mathcal E\subset\mathcal D^G$ be the distinguished space of invariant generalized functions, and let $\mathcal K(\Omega)$ denote the corresponding subspace for a closed conjugation-invariant subset $\Omega$. For $f\in\mathcal K$, let $\widehat f$ be its conjugation average and let $\tau:H_0(G,\mathcal K)\to\mathcal D^G$ be the induced map. Weightless functions and invariant distributions conjecture. The following assertions hold: $\widehat f\in\mathcal E$ for every $f\in\mathcal K$; $\tau$ is an isomorphism

$$H_0(G,\mathcal K)\cong\mathcal E;$$

and

$$\dim H_0(G,\mathcal K(\Omega))=1$$

for every regular semisimple conjugacy class $\Omega\subset G$. These assertions identify the averaged weightless functions with the relevant invariant distributions and prescribe one-dimensional contributions from regular semisimple conjugacy classes; the supplied text gives no evidence of resolution.