Here I am dealing with the difference of triangulated hull and thick hull. Let $\mathcal{D}$ be a triangulated category and $\mathcal{E}\subset\mathcal{D}$ be a collection of objects. The triangulated hull $\mathcal{E}^{\star}$ in $\mathcal{D}$ is the smallest triangulated subcategory of $\mathcal{D}$ containing $\mathcal{E}$, the thick hull $\langle\mathcal{E}\rangle$ in $\mathcal{D}$ is the smallest thick subcategory of $\mathcal{D}$ containing $\mathcal{E}$.

Consider $X$ a complex smooth projective variety of dimension $n$ with a polarization $\mathcal{O}_X(1)$ or say $\mathcal{O}_X(1)$ is very ample, then we know that the thick hull of $\mathcal{E}=\{\mathcal{O}_X,\dots,\mathcal{O}_X(n)\}$ in $D^b(X)$ is $D^b(X)$.

Is $\mathcal{E}^{\star}=D^b(X)$?

I know it is true in the case of $X=\mathbb{P}^n$.