The arithmetic transfer conjecture in the unramified almost self-dual case

Assume that $F/F_0$ is unramified. Let $K'\subset S_n(F_0)$ and $K_1\subset\mathrm{U}(W_1)(F_0)$ be as in the almost self-dual transfer conjecture, let $\omega_S$ be the transfer factor, and let $\operatorname{Int}(g)$ be the relevant intersection number. For a function $f'$ transferring to $(0,\mathbf{1}_{K_1})$, define the corrected orbital-integral relation using a correction function $f'_{\mathrm{corr}}$. Arithmetic transfer conjecture. (a) For every such $f'$, there exists $f'_{\mathrm{corr}}$ such that, for every matching $\gamma\in S_n(F_0)_{\mathrm{rs}}$ and $g\in\mathrm{U}(W_0)(F_0)_{\mathrm{rs}}$,

$$\omega_S(\gamma)\operatorname{\partial Orb}(\gamma,f')=-\operatorname{Int}(g)\cdot\log q+\omega_S(\gamma)\operatorname{Orb}(\gamma,f'_{\mathrm{corr}}).$$

(b) If $\gamma\in S(F_0)_{\mathrm{rs}}$ matches $g\in\mathrm{U}(W_0)(F_0)_{\mathrm{rs}}$, then

$$\omega_S(\gamma)\operatorname{\partial Orb}\bigl(\gamma,(-1)^{n-1}\mathbf{1}_{K'}\bigr)=-\operatorname{Int}(g)\cdot\log q.$$

This conjecture combines a corrected arithmetic transfer identity with an arithmetic fundamental lemma for the distinguished test function.