Explicit unirationality construction for the Klein cubic

Construct a comparatively simple explicit dominant equivariant rational map from a linear representation to the Klein cubic threefold that realizes its $mathsf{PSL}_2(\mathbb{F}_{11})$-unirationality.

Background

The paper proves that the Klein cubic threefold is $mathsf{PSL}_2(\mathbb{F}_{11})$-unirational$$by showing that its action is equivariantly Pfaffian for the central extension$$\mathsf{SL}2(\mathbb{F}{11})$.

Although this establishes the existence of an equivariant unirational parametrization, the authors explain that the resulting explicit map from a linear representation is likely to be extremely complicated. The unresolved question is whether a substantially more natural or “nice” explicit construction exists.