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This is conjectured to hold for varieties with trivial canonical bundle (Bondal-Orlov), and is known to be true for Calabi-Yau 3-folds (Bridgeland established this in his seminal paper https://arxiv.org/abs/math/0009053). In higher dimensions this is one of the basic problems in the field, and, to the best of my knowledge, very little progress has been made towards its solution.