The two-skew-lines conjecture for CI-liaison classes in projective four-space

Let $C$ be a set of two skew lines in $\mathbb{P}^{4}$, spanning a hyperplane $H$, and let $C'$ be another set of two skew lines in $\mathbb{P}^{4}$, spanning a hyperplane $H'$. Two-skew-lines conjecture. The subscheme $C$ is in the CI-liaison class of $C'$ if and only if $H=H'$. This would make the spanning hyperplane a geometric invariant of the CI-liaison class, indicating the existence of an algebraic invariant beyond the Rao module; the conjecture is presented as an open problem in the source.