Vanishing of the regular semisimple obstruction for symmetric pairs

Let $(\operatorname{G},\mathrm{H})$ be a symmetric pair over $k$ such that $\operatorname{G}$ is quasi-split over $k$ and $\operatorname{G}_{\mathrm{der}}$ is simply connected. Set $\X=\mathrm{H}\backslash\operatorname{G}$, and for a regular semisimple point $\xi\in(\X\sslash\mathrm{H})(k)$ let $d_\xi$ denote the natural obstruction class. Obstruction-vanishing conjecture. The natural map

$$[\X/\mathrm{H}](k)\longrightarrow(\X\sslash\mathrm{H})(k)$$

is surjective over the regular locus; equivalently, $d_\xi=0$ there. The paper proves the corresponding equivalence with the existence of a pure inner form when stabilizers are abelian and conjectures this vanishing in the stated generality.