Arithmetic biquadratic fundamental lemma

Let $F$ be the base local field, let $K/K_3$ be unramified, and let

$$\Phi_0:K_0\to M_{2h}(F),\quad \Phi_1:K_1\to M_{2h}(F),\quad \Phi_2:K_2\to M_{2h}(F),\quad \Phi_3:K_3\to M_{2h}(F)$$

be embeddings such that $(\Phi_1,\Phi_2)$ is a regular semisimple pair and $(\Phi_0,\Phi_3)$ is regular semisimple and matching. Let $X$ be the Lubin–Tate formal scheme and $Y_1,Y_2$ the formal schemes of formal $\mathcal O_{K_i}$-modules, with closed immersions $f_i:Y_i\to X$, and let $\mathbf 1$ be the characteristic function of $\mathrm{GL}_{2h}(\mathcal O_F)$. Arithmetic biquadratic fundamental lemma. One has

$$\frac{\pm1}{\log(q^2)}\left.\frac{d}{ds}O_{(\Phi_0,\Phi_3)}(\mathbf 1;s,\eta)\right|_{s=0}=\operatorname{len}_W H^0\left(X,f_{1*}\mathcal O_{Y_1}\otimes_{\mathcal O_X}f_{2*}\mathcal O_{Y_1}\right).$$

This is the arithmetic, or central-derivative, counterpart of the biquadratic fundamental lemma: the derivative of the vanishing central orbital integral is related to an intersection length on Lubin–Tate space.