The COP-Farrell-Jones conjecture for Hecke algebras

Let $G$ be a td-group, meaning a locally compact, second countable, totally disconnected Hausdorff group. Let $R$ be a uniformly regular ring with $\mathbb{Q} \subseteq R$, where uniformly regular means Noetherian and such that some uniform finite bound exists on the lengths of projective resolutions of finitely generated projective $R$-modules. Let $\mathcal{H}(G;R)$ be the Hecke algebra, let $H_n^G(-;\mathbf{K}_R)$ be a smooth $G$-homology theory with $H_n^G(G/U;\mathbf{K}_R) \cong K_n(\mathcal{H}(U;R))$ for every open subgroup $U$, and let $E_{{\mathcal{C}\hspace{-1pt}\mathrm{op}}}(G)$ be the classifying space for proper smooth $G$-actions.

COP-Farrell-Jones conjecture for Hecke algebras. For every uniformly regular ring $R$ with $\mathbb{Q} \subseteq R$, the projection

$$E_{{\mathcal{C}\hspace{-1pt}\mathrm{op}}}(G) \to G/G$$

induces, for every $n \in \mathbb{Z}$, an isomorphism

$$H^G_n(E_{{\mathcal{C}\hspace{-1pt}\mathrm{op}}}(G);\mathbf{K}_R) \xrightarrow{\cong} H^G_n(G/G;\mathbf{K}_R) = K_n(\mathcal{H}(G;R)).$$

This conjecture is stated in the source as a formulation of the $K$-theoretic Farrell-Jones conjecture for Hecke algebras of totally disconnected groups. It is known for reductive $p$-adic groups in the cited forthcoming work, while the general case remains open.