Clozel's formula for higher homology of reductive p-adic groups

Let $G$ be a reductive $p$-adic group. Choose a minimal parabolic subgroup $P_0\subset G$ with Levi decomposition $P_0=L_0U_0$. For each parabolic subgroup $P\supset P_0$ with Levi decomposition $P=LU$ and $L_0\subset L$, write $L\leq G$, let $\operatorname{r}^G_L$ be Jacquet restriction along $P$, and let $\overline{\operatorname{i}}_L^G$ be parabolic induction along the opposite parabolic. Let $\chi_L:L\to\{0,1\}$ be Clozel's characteristic function, let $L_{cz}$ be the union of the compact-mod-centre subgroups of $L$, and let $1_{L_{cz}}$ be its characteristic function. Clozel's higher-homology conjecture. The following is an equality of operators on $\operatorname{H}(\operatorname{H}(G))$:

$$\sum_{L\leq G} \overline{\operatorname{i}}_L^G 1_{L_{cz}}\chi_L \operatorname{r}^G_L=1.$$

This generalizes Clozel's formula from degree zero to higher homology. The formula is proved in the paper for $G=\operatorname{SL}_2(F)$, while the asserted identity for all reductive $p$-adic groups remains open.