Grothendieck conjectured in the sixties that the homological equivalence relation on algebraic cycles coincides with the numerical equivalence relation. In this note we extend this celebrated conjecture from smooth projective schemes to the broad setting of smooth proper dg categories. As an application, we prove that Grothendieck's original conjecture is invariant under homological projective duality. This leads to a proof of Grothendieck's conjecture in the case of intersections of quadrics and intersections of bilinear divisors. Along the way, we prove also the case of quadric fibrations.