Relative fundamental lemma in the semi-Lie case

Let $F$ be the local field and let $S_n(F)$, $V'_n(F)$, $_{\mathrm{rs}}$, $_{\mathrm{rs}}^+$, and $_{\mathrm{rs}}^-$ be the spaces and regular semisimple loci used above. Let $u_n$ be the transfer factor $u_n= u_n(\gamma,\mathbf u,\mathbf v^\top)$ defined in the geometric comparison, let $\operatorname{BC}^{\eta^{n-1}}_{S_n}$ denote the base-change map, and let $\Lambda_n$ be a self-dual lattice of full rank in $\mathbb V_n^+$. For $\phi\in\mathcal H(S_n(F))$ and $(\gamma,\mathbf u,\mathbf v^\top)\in(S_n(F)\times V'_n(F))_{\mathrm{rs}}$, the relative fundamental lemma asserts

Relative fundamental lemma in the semi-Lie case.

$$\nu_n(\gamma,\mathbf u,\mathbf v^\top)\operatorname{Orb}(\phi\otimes\mathbf 1_{O_F^n\times(O_F^n)^\vee},(\gamma,\mathbf u,\mathbf v^\top),0)= \begin{cases} 0&\text{if }(\gamma,\mathbf u,\mathbf v^\top)\in(S_n(F)\times V'_n(F))_{\mathrm{rs}}^-,\\ \operatorname{Orb}^{\operatorname{U}(\mathbb V_n^+)\times\mathbb V_n^+}\bigl((g,u),\operatorname{BC}^{\eta^{n-1}}_{S_n}(\phi)\otimes\mathbf 1_{\Lambda_n}\bigr)&\text{if }(\gamma,\mathbf u,\mathbf v^\top)\in(S_n(F)\times V'_n(F))_{\mathrm{rs}}^+. \end{cases}$$

Here $\operatorname{Orb}$ is the weighted orbital integral defined using the character $\eta$ and the factor $|\det h|_F^{-s}$, while the right-hand side is the ordinary orbital integral for $\operatorname{U}(\mathbb V_n^+)\times\mathbb V_n^+$. This is a semi-Lie relative fundamental lemma relating weighted orbital integrals on the linear side to ordinary unitary orbital integrals after base change; the source gives no resolution, so the conjecture remains open.