The section conjecture for generic pointed curves

Let $k/\mathbb Q$ be finitely generated, let $g\geq3$ and $n\geq1$, and let $\mathcal M_{g,n/k}$ be the moduli stack of smooth proper genus-$g$ curves with $n$ distinct ordered markings. Put $K=k(\mathcal M_{g,n/k})$. We prove Grothendieck's section conjecture for the generic $n$-pointed genus-$g$ c