Transfinite Adams representability

We consider the following problems in a well generated triangulated category TT. Let α be a regular cardinal and Tα⊂TTα⊂T the full subcategory of α  -compact objects. Is every functor H:(Tα)op→AbH:(Tα)op→Ab that preserves products of α≥ℵ2ℵω>α≥ℵ2.