Weighted Langlands–Shelstad transfer conjecture

Let $G$ and $M$ be the groups under consideration over a local field $F$. Let $M'$ be an unramified local elliptic endoscopic datum for $M$, let $\delta'\in\Delta_{G\textnormal{-reg}}(M',\zeta')$, and let $f'$ belong to the relevant test-function space. The transfer is defined by

$$J_{M}(\delta',f)=\sum_{\gamma\in\Gamma_{G\textnormal{-reg}}(M,\zeta)}\Delta_{M}(\delta',\gamma)J_{M}(\gamma,f).$$

Weighted Langlands–Shelstad transfer conjecture. For each $G$ and $M$, there is a function $\hat{S}^{{\tilde G}'}_{{\tilde M}'}(\delta',f')$ on

$${\mathscr S}_{\mathrm{ac}}(\tilde{G}',\tilde\zeta')$$

such that

$$J_{M}(\delta',f)=\sum_{G'\in {\mathscr E}_{M'}(G)}\iota_{M'}(G,G')\hat{S}^{{\tilde G}'}_{{\tilde M}'}(\delta',f').$$

This is the weighted form of the Langlands–Shelstad transfer conjecture, asserting that the transfer of weighted orbital integrals is expressed through the corresponding endoscopic terms. The supplied text does not state whether the conjecture is open or resolved.