Hovey–Palmieri conjectures on Bousfield classes

Let $\mathcal{T}$ be a triangulated category with coproducts. For an object $H\in\mathcal{T}$, its cohomological Bousfield class is $\operatorname{Ker}H^*=\mathopen{\{}t\in\mathcal{T}\mathrel{|}\operatorname{Hom}(t,\Sigma^iH)=0\text{ for all }i\in\mathbb{Z}\mathclose{\}}$. A Bousfield class is a subcategory of the form $\operatorname{Ker}H_*$, and a localizing triangulated subcategory is a triangulated subcategory closed under coproducts.

Hovey–Palmieri conjectures. The following assertions hold:

1. Every cohomological Bousfield class is a Bousfield class.
2. Every localizing tensor ideal is a Bousfield class.
3. Every localizing triangulated subcategory is a Bousfield class.

Any one of these assertions would imply that every localizing tensor ideal of the homotopy category of spectra admits a Bousfield localization. The source presents these as conjectures of Hovey and of Hovey–Palmieri; their resolution status is not established by the supplied text.