Stable germ expansion conjecture for quasi-split reductive groups

Let $G$ be a connected quasi-split reductive group over a non-archimedean local field $F$. Let $\mathcal{SU}_G$ be the vector space of stable distributions on $G(F)$ with true unipotent support, let $G_{reg}(F)$ denote the regular semisimple elements of $G(F)$, and write $\mathcal{SO}_G(t,f)$ for the stable orbital integral of $f\in\mathcal{C}_c^\infty(G(F))$ at $t\in G_{reg}(F)$. For $z\in Z(F)$, where $Z$ is the center of $G$, let $V$ be a neighbourhood of $z$ in $G(F)$. Stable germ expansion conjecture. (1) The vector space $\mathcal{SU}_G$ is finite dimensional. (2) There is a map

$$\mathcal{S}\Gamma_G:G_{reg}(F)\to\mathcal{SU}_G$$

such that: (a) the distribution $\mathcal{S}\Gamma_G(t)$ depends only on the stable conjugacy class of $t\in G_{reg}(F)$; and (b) for every $f\in\mathcal{C}_c^\infty(G(F))$ and $z\in Z(F)$, there is a neighbourhood $V$ of $z$ such that, for $t\in V\cap G_{reg}(F)$,

$$\mathcal{SO}_G(t,f)=\langle\mathcal{S}\Gamma_G(t),f\rangle.$$

This conjecture extends the stated germ-expansion theorem from the previously treated setting to arbitrary connected quasi-split reductive groups over non-archimedean local fields. Its first assertion is known in characteristic zero and expected in general, while the second requires the restriction to true unipotent elements because primitive rational elements can become unipotent only after extending the field; the structure of stable distributions with true unipotent support remains poorly understood.