# Is the injective model structure on symmetric spectra Bousfield localizable?

I am interested in injective model structures on both symmetric spectra as exposed in Hovey/Shipley/Smith and motivic symmetric spectra as in Jardine's article. Both authors take a model structure on the underlying spaces - simplicial sets in Hovey/Shipley/Smith and motivic spaces in Jardine - with monomorphisms as cofibrations. Then both establish two level model structures on symmetric (resp. motivic symmetric) spectra with weak equivalences given levelwise, an injective one with levelwise cofibrations and a projective one with levelwise fibrations.

However, both then proceed using only the projective model structure, and Bousfield localizing it with respect to an appropriate class of maps. Presumably they do so because it is much more straightforward to verify that the projective model structure satisfies the prerequisites for a Bousfield localization, in particular one has easy candidates for generating cofibrations which then turn out to do the job.

My question is: Is the injective level model structure on symmetric spectra (resp. motivic symmetric spectra) known or expected to be cellular or combinatorial? How would you try to prove this?