An old conjecture of Bondal–Orlov–Kawamata predicts that K-equivalent varieties are D-equivalent, see Kawamata's paper D-equivalence and K-equivalence for definitions. In particular this applies to birational Calabi–Yau varieties. See Progress on Bondal–Orlov derived equivalence conjecture for the Bondal–Orlov formulation of the conjecture and the known cases. These conjectures are mostly open, and considered to be an important bridge between derived categories and birational geometry.

Derived categories and Chow motives play the role of universal cohomology theories, in noncommutative, and commutative worlds respectively. Do we expect that *K-equivalence implies isomorphism of rational, or even integral Chow motives*?

**Example 1.** Integral Chow motives of varieties related by a standard flop are isomorphic: Q. Jiang - On the Chow theory of projectivization. This was the motivation for the question.

**Example 2** (from Nico Berger's comment). Birational irreducible holomorphic symplectic varieties have isomorphic integral Chow motives: U. Riess - On the Chow ring of birational irreducible symplectic varieties.

**Remarks.** In many cases, D-equivalence is known to imply isomorphisms for rational Chow motives, but not for integral ones: examples can be found among K3 surfaces. It is also known that D-equivalence *does not* imply K-equivalence, even for rational surfaces: Uehara - An example of Fourier–Mukai partners of minimal elliptic surfaces.

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