Spectral decomposition conjecture for regular elliptic orbital integrals of $PGL(2,F)$

Let $F$ be a local non-Archimedean field and let $G$ be the group of $F$-points of a reductive group, with $\hat G_t$ denoting the set of tempered representations. For a regular elliptic conjugacy class $\Omega\subset G$, let $I_\Omega$ be its orbital-integral functional, and let $\chi_\pi$ denote the character of a representation $\pi\in\hat G$. Spectral decomposition conjecture. For every regular elliptic conjugacy class $\Omega\subset G$, there exists a unique measure $\mu_\Omega$ on $\hat G_t$ such that

$$I_\Omega=\int_{\pi\in\hat G}\chi_\pi\\,\mu_\Omega.$$

This asserts that every regular elliptic orbital integral has a unique spectral decomposition in terms of tempered characters. The supplied text gives no resolution status, so the conjecture is recorded as open.