There is a conjecture of Orlov stating that if $X$ and $Y$ are smooth projective complex varieties that are derived equivalent (equivalent bounded derived categories of coherent sheaves), then their rational Voevodsky motives $M(X)\_{\mathbb{Q}}$ and $M(Y)\_{\mathbb{Q}}$ are equivalent.


My question is whether the equivalence of $M(X)\_{\mathbb{Q}}$ and $M(Y)\_{\mathbb{Q}}$ implies the partial converse that the rational(!) bounded derived categories of coherent sheaves are equivalent? There is a result of Cisinski and Tabuada that implies that the equivalence of the rational Voevodsky motives implies that the noncommutative rational motives of $X$ and $Y$ are equivalent in Kontsevich's category of rational noncommutative motives.



Equivalence of rational Voevodsky motives: partial Converse to Conjecture of Orlov
Definitely not. Take $X$ to be the blowup of $P^2$ at a point and $Y$ to be $P^1 \times P^1$. Then
$$
M(X) = 1 + 2L + L^2 = M(Y),
$$
but the derived categories are different, since both varieties are Fano and non-isomorphic,


