The component-count conjecture for the lifted scattering correspondence

Let $\widetilde{\rm C}(3,n,1)$ be the lifted scattering correspondence over the spinor-helicity variety $X(3,n)$, and let ${\rm C}(3,n,1)$ be the corresponding scattering correspondence. The lifted correspondence comes with a map to $X(3,n)$, and the involution referred to in the source acts on its birational components.

Component-count conjecture. The lifted scattering correspondence $\widetilde{\rm C}(3,n,1)$ decomposes into five irreducible components, with exactly one component whose map onto $X(3,n)$ is not birational. Consequently, the scattering correspondence ${\rm C}(3,n,1)$ has three irreducible components, and the four birational components of the lifted correspondence become two components modulo the involution.

This predicts the irreducible-component structure of the scattering correspondence and distinguishes the unique non-birational component from the birational ones. The supplied text gives no evidence that the conjecture has been proved or disproved.