Nontrivial L-equivalent cubic fourfolds

Construct or determine whether there exist non-isomorphic L-equivalent cubic fourfolds $X$ and $Y$ such that their classes differ in $K_0(\mathrm{Var}_C)$.

Background

The paper derives relations between the Grothendieck classes of the Enriques surfaces and cubic fourfolds related by the Enriques Cremona transformation. It then records a question from the literature concerning the existence of genuinely nontrivial L-equivalent cubic fourfolds: varieties that are L-equivalent but whose Grothendieck classes are not equal.

References

The authors moreover ask whether there are nontrivially $L$-equivalent cubic fourfolds, i.e. $L$-equivalent cubics $X$ and $Y$ such that $[X]\neq[Y]$ in $K_0(\mathrm{Var}_C)$.

— Birational cubic fourfolds via an Enriques Cremona transformation  (2609.10353 - Brooke et al., 9 Sep 2026) in Remark following Corollary \ref{cor: L equivalence}, Section 3