Admissible subcategories of $D^b(\mathbb{P}^n)$

Question

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 $D^b(\mathbb{P}^n_k)$.

Answer 1

Reposting my comment as a slightly extended answer.

For $n=1$ this is folklore (it is I think a pleasant exercise using global dimension 1 and the description of coherent sheaves on curves), whilst for $n=2$ it was settled last year by Pirozhkov in his preprint Admissible subcategories of del Pezzo surfaces. He also gives the analogous statement for del Pezzo surfaces.

For $n\geq 3$ it's still open as far as I know.