Periods of the nilpotent completion of the fundamental group

The nilpotent completion π₁(X)^{^} of the topological fundamental group of a smooth complex  quasi-projective variety X carries a mixed Hodge structure, in particular, it has periods. Some of those have been shown to be multi-zeta values. In general, one does not know what they are. Assuming the variety X itself is highly rational, one can lift π₁(X)^{^} in the category of mixed Tate motives. This restricts the range of possible periods, and also gives some information on relations.