Homological Bondal-Orlov localization conjecture for rational singularities

Given a resolution of rational singularities $\pi \text{\hspace{0.17em}⁣}:\tilde{X}\to X$ over a field of characteristic zero we use a Hodge-theoretic argument to prove that the image of the functor $\mathbf{R}{\pi }_{\ast }\text{\hspace{0.17em}⁣}:\mathbf{D}(\tilde{X})\to \mathbf{D}(X)$ between bounded derived categories of coherent sheaves generates $\mathbf{D}(X)$ as a triangulated category. This gives a weak version of the Bondal-Orlov localization conjecture, answering a question of Pavic and Shinder. The same result is established more generally for proper (non-necessarily birational) morphisms $\pi \text{\hspace{0.17em}⁣}:\tilde{X}\to X$, with $\tilde{X}$ smooth, satisfying $\mathbf{R}{\pi }_{\ast }({\mathcal{O}}_{\tilde{X}})={\mathcal{O}}_X$.