Motivic fundamental lemma for orthogonal Lie algebras

Let $n$, $k$, and $r$ be the parameters indexing the relevant classical groups and equal-valuation strips. Let $\Theta^{G,+}_{n,k,r}$ and $\Theta^{G,-}_{n,k,r}$ be the Chow motives associated with the positive and negative transfer-factor parts, let $\Theta^{H,st}_{n,k,r}$ be the motive associated with the stable endoscopic tube, and let $\mathbb{L}$ denote the Lefschetz motive. Motivic fundamental lemma. Given $n$, $k$, and $r$,

$$\mathbb{L}^c\bigl(\Theta^{G,+}_{n,k,r}-\Theta^{G,-}_{n,k,r}\bigr)=\Theta^{H,st}_{n,k,r}$$

in

$$\widehat K_0^v\bigl(M_{\mathbb{Q}(Z_r),\overline{\mathbb{Q}}}\bigr)_{\mathrm{loc},\mathbb{Q}}.$$

This is the motivic formulation intended to globalize the $p$-adic fundamental lemma through Denef–Loeser motives; the supplied text gives no evidence that it has been proved or refuted.