  In this paper, we discuss the problem of whether the difference $[X]-[Y]$ of
the classes of a Fourier--Mukai pair $(X, Y)$ of smooth projective varieties in
the Grothendieck ring of varieties is annihilated by some power of the class
$\mathbb{L} = [ \mathbb{A}^1 ]$ of the affine line. We give an affirmative
answer for Fourier--Mukai pairs of very general K3 surfaces of degree 12. On
the other hand, we prove that in each dimension greater than one, there exists
an abelian variety such that the difference with its dual is not annihilated by
any power of $\mathbb{L}$, thereby giving a negative answer to the problem. We
also discuss variations of the problem.
