The arithmetic fundamental lemma for regular semisimple elements

Let $F$ be a non-archimedean local field, let $E/F$ be the relevant quadratic extension, and let $S_n(F)$, $U(J_0\oplus 1)(F)$, and the matching relation be as defined above. For a regular semisimple element $\gamma\in S_n(F)_{\rm rs}$, let $O(\gamma,1_{S_n(\mathcal{O}_F)})$ be the weighted orbital integral, let $g$ be a matching regular semisimple element when one exists, let $K$ be the stabilizer of the chosen self-dual lattice, and let $\omega(\gamma)$ be the sign defined by the valuation formula in the paper.

The arithmetic fundamental lemma. For $\gamma\in S_n(F)_{\rm rs}$,

$$O(\gamma,1_{S_n(\mathcal{O}_F)})= \begin{cases} \omega(\gamma)O(g,1_K),&\text{if }\gamma\text{ matches }g\in U(J_0\oplus 1)(F)_{\rm rs},\\\\ 0,&\text{if }\gamma\text{ matches no }g\in U(J_0\oplus 1)(F)_{\rm rs}. \end{cases}$$

This is the minuscule-case fundamental lemma for unitary groups. Equal-characteristic results and the $p$-adic case for sufficiently large, unspecified residue characteristic were known according to the source; the general statement in the paper is presented as a conjecture.