Motivic rigidity conjecture for Severi–Brauer varieties under division-preserving extensions

Let $D$ be a division algebra over $F$ and let $E/F$ be a field extension such that $D_E$ remains a division algebra. Consider the Severi–Brauer varieties of $D$ and their motivic decompositions with coefficients in a finite field. Motivic rigidity conjecture. These motivic decompositions lift over $E$. The result concerns the rigidity of motivic decompositions under field extension; the converse, asserting that the indecomposable motives of the Severi–Brauer varieties of $D_E$ are defined over $F$, is stated to be an open problem.