The fundamental lemma for restricted orbital integrals

Let $(\mathfrak{g},\mathfrak{h})$ be any pair in Definition~, let $r$ be a slope, and let $Y\in\mathfrak{h}$ be regular semisimple. Suppose that the image of $Y$ in $\mathfrak{g}$ is restricted of slope $r$.

The fundamental lemma. The orbital integrals satisfy

$$\operatorname{orb}_{\mathfrak{g}}^{\mathfrak{h}}(Y)=\operatorname{orb}_{\mathfrak{h}}^{\mathfrak{h}}(Y).$$

This is the restricted-slope form of the fundamental lemma, comparing the $\kappa$-orbital integral on $\mathfrak{g}$ with the stable orbital integral on $\mathfrak{h}$. The source does not state a resolution status for this conjecture.