Tensor-product closure of equivariant multiplicativity

Determine whether the tensor product of an equivariantly multiplicative quadratic form for a group G and an equivariantly multiplicative quadratic form for a group H is equivariantly multiplicative for the product group G×H.

Background

The paper establishes that equivariantly multiplicative forms are stably Pfister and proves an inductive construction: if an equivariantly multiplicative form is odd as a function on its vector space, tensoring it with the regular two-dimensional representation of C2 produces an equivariantly multiplicative form for the product group G×C2. The paper also notes that stably Pfister equivariant quadratic forms behave well under tensor product.

What remains unresolved is whether equivariant multiplicativity itself, rather than merely the stable Pfister property, is preserved under arbitrary tensor products of forms associated with two possibly different finite groups. A positive answer would extend the inductive constructions and establish a general product principle for equivariant multiplicative quadratic forms.

References

Suppose that $(V,q)$ and $(W,r)$ are equivariantly multiplicative for $G$ and $H$ respectively. Is $(V\otimes W, q\otimes r)$ equivariantly multiplicative for $G\times H$?

— Witt rings, Pfister forms, and equivariant birational geometry  (2608.17821 - Hassett et al., 18 Aug 2026) in Section “Analysis of unstable Pfister forms,” subsection “Inductive results” (Question at the end of the section)