The pseudo-automorphism group conjecture for the rational threefold X

Let $X$ be the rational threefold, $S$ its Jacobian Kummer surface, $R$ the unique rational normal curve through the six points blown up in constructing $X$, and $PsAut(X)$ the group of pseudo-automorphisms of $X$. Let $G$ be the subgroup generated by the $45$ Hutchinson-Göpel-type involutions and the $120$ Keum pseudo-automorphisms. Pseudo-automorphism group conjecture. For every $g\in PsAut(X)$, $g(R)=R$. In particular, $PsAut(X)=G$. This would fully determine the pseudo-automorphism group of $X$ by showing that every pseudo-automorphism fixes the distinguished curve $R$ and belongs to the subgroup generated by the two explicitly constructed families. The preceding results establish the structure of the subgroup $G$ and a rational polyhedral fundamental domain for its action, but the stated claim is not resolved here.