The relative norm–induction comparison for equivariant commutative ring spectra

Let $G$ be a compact Lie group, let $H<G$ be a closed subgroup, and let $R$ be a cofibrant $H$-equivariant commutative ring orthogonal spectrum in the $U_H$-universe model structure. Relative norm–induction conjecture. There is a natural $U$-universe weak equivalence

$$N_H^G R\simeq R\otimes_H G,$$

where $(-)\otimes_HG$ is left adjoint to the point-set forgetful functor from $G$-equivariant commutative ring orthogonal spectra to $H$-equivariant commutative ring orthogonal spectra. This predicts that the relative norm agrees with the algebraic induction functor after deriving the constructions on cofibrant inputs.