Morava $K(2)$-motives and the Rost invariant

A principal homogeneous bundle under the action of a simple algebraic group determines invariants in the Brauer group called Tits algebras. In the case when the group is simply connected these invariants are trivial, but one can define a higher degree invariant called the Rost invariant. It follows from a result by I. Panin that the motives of full flag varieties with coefficients in the Grothendieck group $K^0$ are isomorphic if and only if the Tits algebras generate the same subgroups in the Brauer group. We propose an analog of this result for the case of the Rost invariant; one considers Morava $K$-theory $K(2)$ instead of $K^0$.