  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 $p=5$.
