Nontriviality of the Enriques moduli involution

Determine whether the rational involution on the moduli space of Fano-polarized Enriques surfaces induced by the Enriques Cremona transformation is nontrivial.

Background

The construction associates to a Fano-polarized Enriques surface S another Enriques surface T and thereby defines a rational involution on the corresponding moduli space. The paper compares S and T through classes in the Grothendieck ring, but does not establish whether the involution actually exchanges distinct moduli points or acts trivially.

References

We have not verified whether this involution is trivial, although we make some preliminary comparisons between the classes of $S$ and $T$ (as well as those of $X$ and $Y$) in the Grothendieck ring of varieties in Corollary~\ref{cor: L equivalence}.

— Birational cubic fourfolds via an Enriques Cremona transformation  (2609.10353 - Brooke et al., 9 Sep 2026) in Section 1, Introduction, remark following Theorem 2