On the Jacobson Element and Generators of the Lie Algebra grt in Nonzero Characteristic

We state a conjecture (due to M. Duflo) analogous to the Kashiwara-Vergne conjecture in the case of a characteristic p > 2, where the role of the Campbell-Hausdorff series is played by the Jacobson element. We prove a simpler version of this conjecture using Vergne's explicit rational solution of the Kashiwara-Vergne problem. Our result is related to the structure of the Grothendieck-Teichmuller Lie algebra grt in characteristic p: we conjecture existence of a generator of grt in degree p - 1, and we provide this generator for p = 3 and p = 5.