The linear arithmetic fundamental lemma for GL$_{2h}$

Let $F$ be a non-Archimedean local field with residue field of cardinality $q$, let $K/F$ be a quadratic étale extension, and let $D$ be a division algebra over $F$ of invariant $\frac{1}{2h}$. Let $(\tau_1,\tau_2)$ be a double $F\times F$-structure on $\mathbf{G}_{2h}(F)=\operatorname{GL}_{2h}(F)$ matching a double $K$-structure on $D$, and let $P_{\tau_1,\tau_2}$ be its invariant polynomial. For a map of $F$-algebras $\alpha:K\to\mathbf{G}_{2h}(F)$ and $g\in\mathbf{G}_{2h}(F)$, let $P_g$ be the invariant polynomial of the double $K$-structure $(\alpha,g^{-1}\circ\alpha\circ g)$. Let $f:\mathbf{G}_{2h}(F)\to\mathbb{R}$ be a spherical Hecke function. The linear arithmetic fundamental lemma. Then

$$\pm(2\ln q)^{-1}\left.\frac{\mathrm{d}}{\mathrm{d}s}\right|_{s=0}\mathrm{Orb}_{\tau_1,\tau_2}(f,s)=\frac{\boldsymbol{\epsilon}_{F,2h}}{\boldsymbol{\epsilon}_{K,h}^2}\int_{\mathbf{G}_{2h}(F)}f(g)\left|\operatorname{Res}(P_{\tau_1,\tau_2},P_g)\right|_F^{-1}\,\mathrm{d}g.$$

Here $\boldsymbol{\epsilon}_{F,2h}$ and $\boldsymbol{\epsilon}_{K,h}$ are the densities of invertible matrices in $\operatorname{Mat}_{2h}(\mathcal{O}_F)$ and $\operatorname{Mat}_h(\mathcal{O}_K)$, respectively, and $\operatorname{Res}$ denotes the resultant of two polynomials. The identity is the conjectural linear arithmetic fundamental lemma, relating the derivative of a relative orbital integral to an integral weighted by the resultant of invariant polynomials; the paper gives a computational proof in the stated setting.