The period-index conjecture is false for motivic reasons

For a Brauer class over the function field of an algebraic variety, the period-index conjecture predicts a bound on one measure of its complexity (the index) in terms of another (the period) and the dimension of the variety. The conjecture holds in dimension two by a famous theorem of de Jong. I will explain why it fails in all higher dimensions due to an obstruction of motivic nature, and propose a corrected conjecture to which this obstruction does not apply.