Recall that a triangulated subcategory $\mathcal{A}$ of a triangulated category $\mathcal{B}$ is called admissible if the inclusion functor has both left and right adjoints.

Is it true that all admissible subcategories of $D^b(\mathbb{P}^n_k)$ (the bounded derived category of coherent sheaves on $\mathbb{P}^n_k$, for a field $k$) are generated by exceptional collections?

Perhaps the right setting for the question is to consider admissible subcategories $\mathcal{A}$ of a $k$-linear triangulated category $\mathcal{B}$ generated by a (strong) exceptional collection rather than just $D^b(\mathbb{P}^n_k)$.