Neeman introduced approximable triangulated categories to formalize the interplay between compact generation and t-structures on triangulated categories with coproducts. This formalism has proved to be a powerful tool for proving conjectures, generalizing Rickard's theorem on derived invariance, and establishing results on the uniqueness of enhancements of geometric categories. In his celebrated paper (Invent. Math. 2026), Neeman asked whether the derived category of a differential graded (dg) algebra with cohomology vanishing in high degrees is weakly approximable or approximable. In this talk, I will discuss the latest progress on this question.