The equivalence conjecture for irreducible finite-dimensional sheaves and pure motives

Let ${\mathbb B}^{\bullet}$ be the fully faithful functor from pure $k$-motives to semisimple sheaves of finite length of finite-dimensional graded ${\mathbb Q}$-vector spaces, with ${\mathbb B}^0(M)$ the sheaf associated to a pure motive $M$. Equivalence conjecture. This is an equivalence of categories; equivalently, every irreducible sheaf of finite-dimensional ${\mathbb Q}$-vector spaces is isomorphic to ${\mathbb B}^0(M)$ for a primitive irreducible effective pure motive $M$. The conjecture asserts essential surjectivity of the motivic realization functor and would classify the irreducible sheaves in this category by pure motives.