Biquadratic linear AFL conjecture for GL(4)

Let $\alpha:(K_1, K_2) \to \operatorname{Mat}_{2h}(F)$ and $\beta:(K_0, K_3) \to \operatorname{Mat}_{2h}(F)$ be matching regular semisimple pairs. Let $\mathcal{H}_{2h}$ denote the space of spherical Hecke functions. Biquadratic linear AFL conjecture. For any spherical Hecke function $f \in \mathcal{H}_{2h}$, one has

$$\operatorname{Orb}(f, \alpha) = \operatorname{Orb}(f, \beta, 0).$$

This is the biquadratic fundamental lemma, a local identity relating orbital integrals for matching regular semisimple pairs. The source describes the result as a conjecture and notes that it proves a special case when both $\mathbf{w}$ and $\mathbf{z}$ are units; the general statement is therefore not established in the source.