Local lattice-count identity for the Jacquet–Rallis fundamental lemma

Let $(a,b)$ be a strongly regular semisimple collection of invariants with $a_i,b_i\in\mathcal{O}_E^{\sigma=(-1)^i}$, allowing arbitrary $b_0\in\mathcal{O}_F$. Let $\Delta_{a,b}$ be the discriminant attached to these invariants, let $\eta_{E/F}$ be the quadratic character associated with $E/F$, and let $M^{\operatorname{loc}}_{i,a,b}$ and $N^{\operatorname{loc}}_{a,b}$ be the local lattice sets defined in the surrounding discussion. Local lattice-count conjecture. One has

$$\sum_{i=0}^{\operatorname{val}_F(\Delta_{a,b})}\eta_{E/F}(\varpi)^i\#M^{\operatorname{loc}}_{i,a,b}=\#N^{\operatorname{loc}}_{a,b}.$$

This identity is the lattice-count formulation of the relevant orbital-integral comparison: the preceding propositions identify the two orbital integrals with the lattice counts, so proving it yields the corresponding parts of the Jacquet–Rallis conjecture. The supplied text does not state whether this intermediate identity is separately resolved.