Exceptional sequences of line bundles on projective bundles

For a vector bundle $\mathcal{E}\to {\mathbb{P}}^{\mathrm{\ell }}$ we investigate exceptional sequences of line bundles on the total space of the projectivisation $X=\mathbb{P}(\mathcal{E})$. In particular, we consider the case of the cotangent bundle of ${\mathbb{P}}^{\mathrm{\ell }}$. If $\mathrm{\ell }=2$, we completely classify the (strong) exceptional sequences and show that any maximal exceptional sequence is full. For general $\mathrm{\ell }$, we prove that the Rouquier dimension of $\mathcal{D}(X)$ equals $\dim X$, thereby confirming a conjecture of Orlov.