On the Jacobson element and generators of the Lie algebra $\mathfrak{grt}$ 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--Teichm\"{u}ller Lie algebra $\mathfrak{grt}$ in characteristic $p$: we conjecture existence of a generator of $\mathfrak{grt}$ in degree $p-1$, and we provide this generator for $p=3$ and $p=5$.