A going-down theorem for Chow-Grothendieck motives

Let (M(X), p) be a direct summand of the motive associated with a geometrically split, geometrically irreducible variety over a field F satisfying the nilpotence principle. We show that under some conditions, if (M(XE), pE) is a direct summand of another motive ME over a field extension E, then (M(X), p) is a direct summand of M over F.