Let $T$ be a small triangulated category. Under which conditions there exists a triangulated category $B$ closed with respect to (small) coproducts such that $T$ fully embedds into the subcategory of compact objects of $B$ (I don't need this embedding to be an equivalence; yet if some $B$ of this sort exists then one can always "shrink" it so that its subcategory of compact objects will become the Karoubi envelope of $T$)?
I would conjecture that such a category $B$ exists for any $T$; yet does there exist any way to prove this? What is the largest class of small triangulated categories for which this statement is known?
I suspect that the proof should be "known" whenever some "enhancement" exists for $T$; yet any comments or references would be very welcome!