Conjectural trace formula for spherical varieties

Let $I(f)$ be the trace-formula distribution for a test function $f$, and let $I_{geom}(f)=m_{geom}(\theta_f)$ be its geometric expansion. Define $I_{spec}(f)$ by the cuspidal spectral sum when $(G,H)$ is not tempered and by the tempered spectral integral when $(G,H)$ is tempered. Trace formula conjecture. The following assertions hold: (1) when $(G,H)$ is tempered,

$$I_{geom}(f)=I(f)=I_{spec}(f)$$

for all $f\in {\mathcal {C}}_{scusp}(G(F),\chi)$; and (2) when $(G,H)$ is not tempered, the same equality holds for all $f\in {}^\circ {\mathcal {C}}(G(F),\chi)\cap C_{c}^{\infty}(G(F),\chi)$. The conjecture is known in the Whittaker, Gan–Gross–Prasad, Ginzburg–Rallis, Galois, and Shalika models, while it remains open in general.