The stable character formula for supercuspidal Langlands parameters

Let $F$ be a non-archimedean local field, let $G$ be a reductive group that splits over a tame extension of $F$, and assume that the residue characteristic is sufficiently large for the exponential map on topologically nilpotent elements. Let $\varphi:W_F\to {^LG}$ be a supercuspidal parameter, and let $(S,\theta)$ be the associated torus and genuine character, with $a$-data $a_S$. For a strongly regular semisimple element $\gamma\in G(F)$, write its topological Jordan decomposition as

$$\gamma=\gamma_0\cdot\gamma_{>0}.$$

Let $J$ be the connected centralizer of $\gamma_0$ in $G$; let $e(J)$ be the Kottwitz sign of $J$; let $\epsilon(T_G-T_J)$ be the root number of the virtual Galois representation $X^*(T_G)_\mathbb{C}-X^*(T_J)_\mathbb{C}$, where $T_G$ and $T_J$ are minimal Levi subgroups in the quasi-split inner forms of $G$ and $J$; and let the sum range over stable classes of admissible embeddings $S\to J$. Stable character formula. The stable character of the $L$-packet associated with $\varphi$ satisfies

$$S\Theta_\varphi^G(\gamma)=e(J)\epsilon(T_G-T_J)\sum_{j:S\to J}[a_S\cdot\theta](\gamma_0)\cdot\widehat{SO}^J_{^jX^*}(\log(\gamma_{>0})).$$

This formula is intended to give a spectral characterization of the local Langlands correspondence for supercuspidal parameters and, together with endoscopic transfer, to determine the individual members of the associated $L$-packet. The parser supplies no evidence that the assertion has been proved or disproved; its status is therefore open.