The Linear AFL conjecture

Let $M$ be the Rapoport–Zink space of strict $\pi$-divisible $O_F$-modules with quasi-isogeny to $\mathbb{X}$, and let $Z_1,Z_2\to M$ be the closed immersions associated with the matching pair $\beta=(\beta_1,\beta_2)$. Assume that $\beta$ is regular semi-simple, let $L$ be the étale $F$-algebra centralizing the image of $\beta_1(E)\cup\beta_2(E)$, and let $\Gamma\subset L^\times$ be the subgroup generated by chosen uniformizers of the field factors of $L$. Define

$$\operatorname{Int}(\beta)=\ell_{O_{\breve F}}\bigl(\mathcal{O}_{\Gamma\backslash(Z_1\cap Z_2)}\bigr).$$

Let $\alpha=(\alpha_1,\alpha_2)$ be the matching pair associated with $\beta$, and let $O(\alpha,f,s)$ be the corresponding orbital integral. Linear AFL conjecture. Assuming $\alpha$ and $\beta$ are as above, there is an equality up to sign,

$$\left.\frac{\pm 1}{2\log(q_F)}\frac{d}{ds}\right|_{s=0}O\bigl(\alpha,1_{GL_{2n}(O_F)},s\bigr)=\operatorname{Int}(\beta).$$

This is a non-basic case of the arithmetic fundamental lemma, relating the derivative of a central orbital integral to the intersection multiplicity of cycles on a Rapoport–Zink space. The supplied source gives no evidence that the conjecture has been resolved.