Tambara-ideal structure of levelwise Jacobson radicals

Determine whether the levelwise Jacobson radicals of the rings constituting an arbitrary Tambara functor form a Tambara ideal, specifically whether they are closed under transfer, norm, and restriction operations.

Background

The paper defines the Jacobson radical of a Tambara functor as the intersection of its maximal Tambara ideals, rather than by taking the Jacobson radical separately at each level. For multiplicatively cohomological Tambara functors, the authors prove that this Tambara-ideal Jacobson radical agrees levelwise with the ordinary Jacobson radical of each ring.

The authors explain that defining the Jacobson radical levelwise would be useful but may fail to produce a Tambara ideal. Whether the levelwise radicals are preserved by the Tambara operations is therefore left unresolved.