The fixed-cocharacter conjecture for strong involutions

Let $G$ be a reductive group, let $\delta_0$ be the specified automorphism, and let $\xi$ be a strong involution for $G$ with dominant regular integral infinitesimal cocharacter $g$. Assume that the involution $\delta_0\circ\theta_\xi\circ\delta_0$ is equivalent to $\theta_\xi$, meaning conjugate by $G$. Fixed-cocharacter conjecture. There is a $\delta_0$-fixed regular integral cocharacter $g'$ and a strong involution $\xi'$ with infinitesimal cocharacter $g'$ such that $\theta_\xi=\theta_{\xi'}$. The statement is false, so the conjecture has been refuted.