The principal-localizing-subcategory conjecture

A localizing subcategory of the stable homotopy category is a thick subcategory closed under coproducts. For a spectrum $E$, its collection of acyclics consists of spectra $X$ with $E\wedge X=0$.

Principal-localizing-subcategory conjecture. Every localizing subcategory is the collection of $E$-acyclics for some spectrum $E$, and is therefore principal.

The source calls this the basic conjecture about localizing subcategories and notes that, together with Ohkawa's result, it would imply that there is only a set of localizing subcategories. No resolution is given.