Naive matching-function conjecture for parahoric subgroups

Let $G$ be an arbitrary reductive group, let $G^*$ be its quasisplit inner form, and let $J\subset G$ be an arbitrary parahoric subgroup. Put $L=J/J^+$ and let $W_L$ be the Weyl group of $L$. For each $w\in W_L$, let $M_w^*$ be a Levi subgroup of $G^*$ containing the associated unramified maximal torus as an elliptic subtorus, and let $f_{S_{M_w^*}}$ be the relative Euler–Poincaré function defined from representatives of facets in the Bruhat–Tits building. Naive matching-function conjecture. A matching function, in the sense of standard endoscopy, for $1_J$ is

$$\frac{1}{|W_L|}\sum\limits_{w\in W_L} f_{S_{M_w^*}}.$$

This proposes an explicit Iwahori-biinvariant transfer of the parahoric characteristic function to the quasisplit inner form. The statement is formulated as a naive conjecture for general reductive groups, and no resolution is supplied in the source.