$\mathbb Z$-version of the abelian section conjecture

I will describe the proof of the following result:
Theorem. Assume that $K=\bar F_p(X),L=\bar F_l(Y)$ where $X,Y$ are algebraic varieties over $\bar F_p, \bar F_l$ respectively. Let $\psi : K^*/\bar F_p^*\to L^*/\bar F_l$ be homormorphism between the quotients of multiplicative groups. We make the following assumptions:
1) $ dim X \geq 2, dim Y \geq 2$
2)The images of algebraically dependent elements are algebraically dependent.
3) There are at least two elements $x,y\in K^*/ \bar F_p$ such that $\psi(x),\psi(y)$ are algebraically independent
4) $\psi$ has a nontrivial kernel.
Then there is nonarhemedean valuation $\nu$ on $K$ such that the map $\psi$ on the unit subgroup $A_{\nu}^*$ is obtained from a composition of natural projection of the valuation ring $A_{\nu}\to K_{\nu}$ where $K_{\nu}$ is residue field of $\nu$ and a monomorphism $K_{\nu}^*/\bar F_p\to L^*/\bar F_l$.
I will also explain the relation of this result to abelian version of Grothendieck section conjecture.