In the paper "Locally finite triangulated categories" Xiao and Zhu define locally finite triangulated categories. These are triangulated categories $\mathcal{A}$ such that $$\sum_{X \in \mathrm{ind}\mathcal{A}} \mathrm{dim}_k \mathrm{Hom}(X,Y) < \infty$$ for all indecomposable object $Y \in \mathcal{A}$.

Is the (bounded) derived category $D^b(X)$ of a smooth projective scheme $X$ locally finite?

EDIT: In another paper (Han: A note on generic objects and triangulated categories) a triangulated category is defined to be locally finite if for each indecomposable object $X$, there are only finitely many isoclasses of indecomposable objects $Y$ such that $\mathrm{Hom}(X,Y)\neq 0$.