Conceptual explanation for equivariant Pfaffian representations

Identify a conceptual reason why every smooth member of the pencil generated by the Klein and Segre cubic threefolds should be equivariantly Pfaffian for the binary icosahedral group extending $\mathfrak A_5$.

Background

The paper proves that all smooth members of the pencil generated by the Klein and Segre cubic threefolds are $\mathfrak A_5$-unirational by applying a general result for finite subgroups of $\operatorname{PGL}_2$.

The authors observe that an alternative proof would likely follow if every smooth member were equivariantly Pfaffian for the binary icosahedral group extending $\mathfrak A_5$. They regard this equivariant Pfaffian property as likely but explicitly state that they do not know a conceptual explanation for it.